Abstract:
State of the art was the European patent application EP 99105884.3 (see application data sheet). This patent application used already non-linear systems of equations and conditional probabilities with one single item in the condition. It was necessary, however, to perfect these theoretical methods and make them practicable. Many improvements and innovative modifications were needed. The following list identifies the innovations that had to be provided:
       The accurate indication of all systems of equations concerning 2, 3 and 4 hypotheses.   Those equations can be entered in exactly the presented form into the calculation program.   The introduction of coefficients a ik  and b ik  that can be applied without changes for any areas of use.   The delivery of a scheme that enables the mechanized production of the a ik  and b ik .   Introducing schematic tables with identical follow events in one row (e.g. Table 3).   Using appreciation factors AF(i) if the hypotheses K i ′ have the same a-priori probability.   Uncomplicated approach to the causes of the causative events K i  and to the inhibitors.   New factors f ij  for creating a simplification in order to allow an immediate consideration of symptoms which, although expected, did not occur.   Continuous updating of probabilities used.   No self-developed iterative methods of solution are used, but commercially available calculation programs.   A complete and workable example of the automated analysis of electrocardiograms is presented which may serve also as a design template.   The entire operation (using the calculation program selected at this point) is done with just one mouse click.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    See application data sheet. 
       STATEMENT REGARDING FEDERALLY SPONSORING 
       [0002]    N/A 
       THE NAMES OF THE PARTIES TO A JOINT RESEARCH 
       [0003]    N/A 
       REFERENCE TO A COMPACT DISC 
       [0004]    N/A 
       BACKGROUND OF THE INVENTION 
       [0005]    According to the U.S. classification definitions, the invention belongs to Class Number 706, Data processing: artificial intelligence. It belongs as well to Class Number 708, Electrical computers: arithmetic processing and calculating. This is due to the universal usability of the invention, e.g. as a means to evaluate a patient&#39;s electrocardiogram or as a means to survey offshore wind powered electricity generators. 
         [0006]    By using the method it becomes possible to calculate precisely the appreciation factors AF(i) or the a-posteriori probabilities x i  of 2, 3 or 4 hypothetical diagnoses. Required are conditional probabilities which fulfill the basic structure p(following event|causative event). It is emphasized that these conditional probabilities have only one element in the condition. The invention is equipped with a complete and workable example of a calculation program which may be used immediately and without modifications to evaluate electrocardiograms. The attached example is the clear proof that the method works well, sure, and according to mathematical principles. 
       BRIEF SUMMARY OF THE INVENTION 
       [0007]    The invention represents a universal method that can be used in numerous fields of knowledge, for example, earthquake research, geological prospecting, criminal forensics, aircraft accident investigation, on-board diagnosis in road cars and aircraft, monitoring of sea-based wind turbines and the entire field of medicine. In the latter case it is the first task to analyze electrocardiograms. 
         [0008]    The procedure under discussion is applicable in all cases where several hypotheses stand for selection and the most likely candidate will be determined by the symptoms observed and the surrounding hypotheses. Therefore an algebraic method had to be chosen to take into account not only the symptoms but also the competing hypotheses and—according to need—the inhibitors. On mathematical basis, using the Discrete Stochastic, a method was designed to establish the systems of equations required to determine the unknown probabilities of the hypotheses. 
         [0009]    The computer used performs the calculations, and as input it receives in the simplest and most practical case the symptoms observed or the symptoms missing, and as output it provides a list of the proposed diagnoses which are sorted according to their respective probability of existence. 
       BRIEF DESCRIPTION OF THE DRAWINGS 
       [0010]    N/A 
     
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0011]    Causal Structure 
         [0012]    Until now it has been completely impossible during a diagnostic process to adequately consider 1) the symptoms observed, 2) the expected but non-appearing symptoms, 3) the mutual influencing of the diagnoses upon one another, and 4) the a-priori probabilities of the possible diagnoses. In order to accomplish this a technical aid is urgently needed. It is exactly this function which is fulfilled by the presented concept of a diagnostic machine which has the potential to be applied in diverse spheres of activity, and particularly in medicine. 
         [0013]    For medical diagnostics, all diseases of an organ or part of the organ are registered and compiled with their respective symptoms. The disease to be diagnosed is selected and given the code K 1 ′, while any relevant differential diagnoses are encoded as K 2 ′, K 3 ′ and K 4 ′. In this way K′-groups are formed with four competitors, all of which are entered in the same way into the computer, as shown below with a detailed example of a K′ “foursome”. 
         [0014]    The follow events generated from each of the K′-elements form the sets of F-elements; as such (F 11 , . . . , F 16 ) belong to K 1 ′, (F 21 , . . . , F 26 ) are attributable to K 2 ′, (F 31 , . . . , F 36 ) result from K 3 ′ and (F 41 , . . . , F 46 ) belong to K 4 ′, whereby even though the individual F-elements may be indexed differently, in many cases they may actually designate the same symptoms. In general the index i contained in F ij  refers to the causative elements K 1 ′, i: =1, . . . , 4 while the index j denotes the sequential order of the symptoms, with j:=1, . . . , 6. 
         [0015]    As an example, in the following tables 1, 2 and 3 an arbitrary causal structure is shown for K 1 ′. Such a causal structure is calculated, i.e. for all listed K′-elements the a-posteriori probability is determined from the respective a-priori probability. 
         [0016]    The a-posteriori probability of an arbitrarily singled out K 1 ′ is influenced by the F 1j  events and by the competitors (K 2 ′, K 3 ′, K 4 ′) since these competitors can also produce some of the F 1j . The symptom set determined in a case to be diagnosed defines which of the F-elements enter the calculations as negated or non-negated. The negated F-elements represent the previously mentioned symptoms that were expected for K 1 ′, but which actually could not be observed. 
         [0017]    The mathematical actions presented in detail for K 1 ′ must be carried out in the same way for all the other K′-elements whereby the quantity of diagnoses marked by the superscript dash, i.e. (K 1 ′, K 2 ′, K 3 ′, K 4 ′), always remains the same. If K 5 ′ now becomes another differential diagnosis, the negated K 5  is then included in the K′ “foursome” and treated as such, i.e. the presence of K 5  will initially be excluded. In order to confirm the calculation result, each of such excluded diagnoses can then be raised to a primary diagnosis and provided with its own K′-grouping so that it can be calculated in the same way. 
         [0000]    
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Example of a causal structure to calculate K 1 ′. The table 
               
               
                 contains the F 1j -elements which are caused by K 1 ′. It also 
               
               
                 contains the K′-elements which may compete against K 1 ′ 
               
               
                 with respect to the production of the F 1j -elements. 
               
             
          
           
               
                   
                 K 1 ′ 
                 K 2 ′ 
                 K 3 ′ 
                 K 4 ′ 
               
               
                   
                   
               
             
          
           
               
                 F 11   
                 + 
                 + 
                   
                 + 
               
               
                 F 12   
                 + 
                 + 
                 + 
                 + 
               
               
                 F 13   
                 + 
                 + 
                   
                 + 
               
               
                 F 14   
                 + 
                   
                 + 
                 + 
               
               
                 F 15   
                 + 
                   
                 + 
               
               
                 F 16   
                 + 
                 + 
                 + 
               
               
                   
               
               
                 “+” indicates a causal connection between the K′-elements at the top and the F 1j -elements on the left hand side. 
               
             
          
         
       
     
         [0018]    The Table 1 can be rearranged: 

 
         [0019]    The structure shown in Table 1 and 2 is consistent with the implied structure given in Table 3 below. 
         [0020]    {K 1 ′, K 2 ′, K 3 ′, K 4 ′}contain the sought diagnoses (K stands for “Known Competitor”). The elements have an unknown probability of existence (0&lt;p&lt;1) and for this reason bear a superscript dash. The number of competitors is restricted for convenience to four. If the number is increased by one additional ′-element, the length of the calculation equations for the sought unknowns is doubled. 
         [0021]    {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 } contains the symptoms of K 1 ′ (F stands for “Follow Event”). The number of the considered follow events can be freely selected and has been set here arbitrarily to six elements. The events from {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 } are entered into the structure with the probability (p=1), but will change to (p=0) if the symptom set identifies them with this probability. In addition, any event should be removed from the set {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 } if the following criterion is not met for this event: Each of the elements from the set {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 }, which are induced by K 1 ′, should have at least one additional cause from the set {K 2 ′, K 3 ′, K 4 ′}. 
         [0022]    Because of the last-mentioned criterion, it certainly makes sense to create a tabular summary for K 1 ′ and the F ij -symptoms belonging to it, supplemented by the total number of symptoms to be considered, namely S1, . . . , S9. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Tabular overview and continuation of Table 1. The symptoms S1, . . . , S9 are arbitrarily 
               
               
                 chosen. The numerical values of the p(F ij |K i ~) are estimated values. 
               
               
                 The tilde symbol denotes a product of events (synonymous: compound of events, logic 
               
               
                 product) which apart from the K i  entered before the tilde contains all competing 
               
               
                 diagnoses in negated form. 
               
             
          
           
               
                   
                 Diagnoses 
               
             
          
           
               
                   
                 K 1 : Acute 
                 K 2 : Dilated 
                 K 3 : Coronary 
                 K 4 : Toxic 
               
               
                   
                 myocarditis 
                 cardiomyopathy 
                 heart disease 
                 cardiomyopathy 
               
               
                 Symptoms 
                 p(F 1j |K 1 ~) 
                 p(F 2j |K 2 ~) 
                 p(F 3j |K 3 ~) 
                 p(F 4j |K 4 ~) 
               
               
                   
               
             
          
           
               
                 S1: QRS complex 
                 F 11   
                 0.6 
                 F 21   
                 0.6 
                   
                   
                 F 41   
                 0.5 
               
               
                 &gt;0.12 sec 
               
               
                 S2: S waves, low 
                 F 12   
                 0.5 
                 F 22   
                 0.5 
                 F 31   
                 0.3 
                 F 42   
                 0.3 
               
               
                 in V1 + V2 
               
               
                 S3: QRS complex 
                 F 13   
                 0.5 
                 F 23   
                 0.5 
                   
                   
                 F 43   
                 0.3 
               
               
                 split in V5 + V6 
               
               
                 S4: PQ time 
                 F 14   
                 0.6 
                   
                   
                 F 32   
                 0.2 
                 F 44   
                 0.5 
               
               
                 &gt;0.1 sec 
               
               
                 S5: RR intervals 
                 F 15   
                 0.9 
                   
                   
                 F 33   
                 0.5 
               
               
                 varving 
               
               
                 S6: Heart rate 
                 F 16   
                 0.7 
                 F 24   
                 0.4 
                 F 34   
                 0.4 
               
               
                 &gt;100/min 
               
               
                 S7: P-wave in 
                   
                   
                 F 25   
                 0.6 
                 F 35   
                 0.6 
               
               
                 sawtooth shape 
               
               
                 S8: ST-segment 
                   
                   
                 F 26   
                 0.7 
                   
                   
                 F 45   
                 0.2 
               
               
                 lowered 
               
               
                 S9: T-negativity 
                   
                   
                   
                   
                 F 36   
                 0.7 
                 F 46   
                 0.4 
               
               
                 in V2 
               
               
                   
               
             
          
         
       
     
         [0023]    Design of Table 3 
         [0024]    First, the numerical values of the probabilities p(F ij |K i ˜) are statistically determined. For the purpose of determining these values, the i-indexing is the same for F and K. 
         [0025]    To achieve an orderly procedure, all symptoms to be considered are recorded in the first column. For each diagnosis, an additional column is then reserved. All follow events (symptoms) of a diagnosis are entered in the column reserved for this diagnosis, along with their p(F ij |K i ˜) numerical values, the i-indexing for F and K being the same. The ordering is to be carried out in such a way that identical follow events, i.e. events with different F ij  indexing but the same symptom affiliation, are positioned in one row. 
         [0026]    The numerical value, e.g. for p(F 26 |K 4 ˜), can then be read off easily as zero, when nothing is entered at the intersection of the F 26  row and the K 4  column, or as the numerical value which has already been determined and entered for a follow event affiliated to K 4 . If for instance the numerical value for p(F 45 |K 4 ˜) is entered at the intersection of the F 26  row and the K 4  column, this value is then assumed for p(F 26 |K 4 ˜), since although the two follow events F 26  and F 45  are indexed differently, they actually refer to the same symptom in the first column of the table. This symptom in this example is “ST-segment lowered”, which in this case is designated as S8. 
         [0027]    Note 1 
         [0028]    Table 1, 2 and 3 provide the basic structure for the calculation example at the end of the presentation. This calculation example can be used as a model framework for creating tools to make diagnostic decisions for numerous tasks in the field of medicine and outside medicine as well. 
       Requirements 
       [0000]    
       
         
           
             The elements in {K 1 ′, K 2 ′, K 3 ′, K 4 ′} are stochastically independent. 
             The elements in {K 1 ′, K 2 ′, K 3 ′, K 4 ′} are stochastically self-reliant causes (this means that the inhibitors of the causal pathway leading away from the K′-elements are stochastically independent and in addition they are stochastically independent with regard to the K′-elements). 
             {K 1 ′, K 2 ′, K 3 ′, K 4 ′} contains all events that are the cause of two or more elements from {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 }. 
           
         
       
     
         [0032]    Comments on the Requirements 
         [0033]    For any two elements from {F 11 , F 12 , F 13 , F 14 , F 15 , F 16 } conditional stochastic independence can be achieved if the condition contains all causes which the two events have in common. If that is not the case, and if e.g. K 5 ′ is another cause of at least two elements from the set {F 11 . F 12 , F 13 , F 14 , F 15 , F 16 }, the absence of K 5 ′ is assumed and the negated K is then included in the causal structure and in the calculations (whereby a conceptual inclusion is sufficient). 
         [0034]    It is also important that conditional stochastic independence will only be achieved if the causes in the condition constellation are attributed (p=1) or (p=0), but not (0&lt;p&lt;1), i.e. they should not have a superscript dash. 
         [0035]    Regarding the subsequent use of inhibitors, the requirement of self-reliant causes is already dealt with here, in that the causal pathways leading from one cause to two follow events should show no interference with one another. This means that the inhibitors which act on such causal pathways must be stochastically independent. 
         [0036]    Four Diagnoses 
         [0037]    Calculation of x 1    
         [0038]    The evaluation environment for K 1 ′ (in brief: W(K 1 ′)) describes a constellation of events containing those elements of the causal structure that influence the probability of existence of K 1 . (W stands for “World of K 1 ′”) In the specified causal structure, W(K 1 ′) consists of the two logic products (F 11  F 12  F 13  F 14  F 15  F 16 ) and (K 2 ′ K 3 ′ K 4 ′), connected by a Boolean “and”. 
         [0039]    The probability of existence for K 1  under the condition of the evaluation environment for K 1 ′, i.e. p(K 1 |W(K 1 ′)), is the sought unknown x 1 . For historical reasons, p(K 1 |W(K 1 ′)) is given the shorter form p(K 1 |K 1 ′). The same applies for x 2 , x 3  and x 4 , From this we get: 
         [0000]        x   1   :=p ( K   1   |W ( K   1 ′)):= p ( K   1   |K   1 ′):= p ( K   1   |F   11    . . . F   16   K   2   ′K   3   ′K   4 ).
 
         [0000]        x   2   :=p ( K   2   |W ( K   2 ′)):= p ( K   2   |K   2 ′):= p ( K   2   |F   21    . . . F   26   K   1   ′K   3   ′K   4 ).
 
         [0000]        x   3   :=p ( K   3   |W ( K   3 ′)):= p ( K   3   |K   3 ′):= p ( K   3   |F   31    . . . F   36   K   2   ′K   1   ′K   4 ).
 
         [0000]        x   4   :=p ( K   4   |W ( K   4 ′)):= p ( K   4   |K   4 ′):= p ( K   4   |F   41    . . . F   46   K   2   ′K   3   ′K   1 ).
 
         [0040]    The unknown x 1 :=p(K 1 |F 11  . . . F 16  K 2 ′ K 3 ′ K 4 ′) is transformed as follows: 
         [0000]    
       
         
           
             
               x 
               1 
             
             := 
             
               
                 p 
                  
                 
                   ( 
                   
                     
                       K 
                       1 
                     
                     | 
                     
                       
                         F 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         F 
                         16 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           K 
                           1 
                         
                          
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                          
                         
                           K 
                           2 
                           ′ 
                         
                          
                         
                           K 
                           3 
                           ′ 
                         
                          
                         
                           K 
                           4 
                           ′ 
                         
                       
                       ) 
                     
                   
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             K 
                             1 
                           
                            
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                            
                           
                             K 
                             2 
                             ′ 
                           
                            
                           
                             K 
                             3 
                             ′ 
                           
                            
                           
                             K 
                             4 
                             ′ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               K 
                               _ 
                             
                             1 
                           
                            
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                            
                           
                             K 
                             2 
                             ′ 
                           
                            
                           
                             K 
                             3 
                             ′ 
                           
                            
                           
                             K 
                             4 
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     1 
                     
                       1 
                       + 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                             ) 
                           
                         
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                                
                               
                                 K 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         1 
                         + 
                         
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       11 
                                     
                                      
                                     
                                         
                                     
                                      
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     
                                       F 
                                       16 
                                     
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       K 
                                       2 
                                       ′ 
                                     
                                      
                                     
                                       K 
                                       3 
                                       ′ 
                                     
                                      
                                     
                                       K 
                                       4 
                                       ′ 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     3 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       11 
                                     
                                      
                                     
                                         
                                     
                                      
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     
                                       F 
                                       16 
                                     
                                   
                                   | 
                                   
                                     
                                       K 
                                       1 
                                     
                                      
                                     
                                       K 
                                       2 
                                       ′ 
                                     
                                      
                                     
                                       K 
                                       3 
                                       ′ 
                                     
                                      
                                     
                                       K 
                                       4 
                                       ′ 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     3 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           + 
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         11 
                                       
                                        
                                       
                                           
                                       
                                        
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       
                                         F 
                                         16 
                                       
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         3 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         4 
                                         ′ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                     | 
                                     
                                       
                                         K 
                                         2 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         3 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         11 
                                       
                                        
                                       
                                           
                                       
                                        
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       
                                         F 
                                         16 
                                       
                                     
                                     | 
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         3 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         4 
                                         ′ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       K 
                                       1 
                                     
                                     | 
                                     
                                       
                                         K 
                                         2 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         3 
                                         ′ 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         ( 
                         
                           independency 
                           , 
                           
                             K 
                             - 
                             elements 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 1 
                 
                   1 
                   + 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         p 
                          
                         
                           ( 
                           
                             
                               K 
                               _ 
                             
                             1 
                           
                           ) 
                         
                       
                     
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 K 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         p 
                          
                         
                           ( 
                           
                             K 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               . 
             
           
         
       
     
         [0041]    The mathematical term in the numerator, Z 1 :=p(F 1  . . . F 6 |  K   1 K 2 ′K 3 ′K 4 ′), is subjected to linear interpolation: 
         [0000]    
       
         
           
             
               p 
                
               
                 ( 
                 
                   
                     
                       F 
                       11 
                     
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                       F 
                       16 
                     
                   
                    
                   
                     
                       
                         K 
                         _ 
                       
                       1 
                     
                      
                     
                       K 
                       2 
                       ′ 
                     
                      
                     
                       K 
                       3 
                       ′ 
                     
                      
                     
                       K 
                       4 
                       ′ 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           K 
                           2 
                         
                          
                         
                           K 
                           3 
                         
                          
                         
                           K 
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           K 
                           2 
                         
                          
                         
                           K 
                           3 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           K 
                           2 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           3 
                         
                          
                         
                           K 
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           K 
                           2 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           3 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           2 
                         
                          
                         
                           K 
                           3 
                         
                          
                         
                           K 
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           2 
                         
                          
                         
                           K 
                           3 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           2 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           3 
                         
                          
                         
                           K 
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         K 
                         4 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                       
                        
                       
                         
                           
                             K 
                             _ 
                           
                           1 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           2 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           3 
                         
                          
                         
                           
                             K 
                             _ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         2 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         
                           K 
                           _ 
                         
                         3 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             K 
                             _ 
                           
                           4 
                         
                          
                         
                           K 
                           4 
                           ′ 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0042]    The linear interpolation for the denominator term N 1 :=p(F 11  . . . F 16 |K 1 K 2 ′K 3 ′K 4 ′) is carried out in the same way, only  K   1  is replaced by K 1 . 
         [0043]    A simplified form of notation is introduced. Defining examples are provided by 
         [0000]        p ( F   11 |˜):= p ( F   11   |  K     1     K     2     K     3     K     4 ) and
 
         [0000]        p ( F   11   |K   1 ˜):= p ( F   11   |K   1     K     2     K     3     K     4 )
 
         [0000]    where the tilde symbol denotes a product of events (synonymous: compound of events, logic product) which apart from the K i  entered before the tilde contains all competing diagnoses in negated form (see earlier definition in Table 3). 
         [0044]    The conditional probabilities, which arise in such interpolations of Z i  and N i , are designated by a ik  and b ik , k:=0, . . . , 7 namely a ik  with the interpolations of Z i  and b ik  with the interpolations of N i . Thus, for example, the first factor appearing after interpolation of Z 1  is replaced by a 10  with 
         [0000]        a   10   :=p ( F   11    . . . F   16   |  K     1   K   2   K   3   K   4 ). 
         [0045]    A factorization with respect to the F-elements then follows: 
         [0000]        a   10   :=p ( F   11   |  K     1   K   2   K   3   K   4 )· . . . · p ( F   16   |  K     1   K   2   K   3   K   4 ).
 
         [0046]    The factorization is made possible by the fact that the logic product of events (  K   1 K 2 K 3 K 4 ) contains all causes of {F 11 , . . . , F 16 }, so that the conditional stochastic independence of the F-elements is reached. 
         [0047]    Note 2 The factorization with respect to the F-elements takes place after the interpolation. 
         [0048]    Next, the factorization with respect to the K-elements takes place, e.g. the factorization of expressions of the form p(F 11 |  K   1 K 2 K 3 K 4 ). To achieve this, the theorem p(  F   11 |  K   1 K 2 K 3 K 4 ):=p(  F   11 |K 2 ˜)·p(  F   11 |K 3 ˜)·p(  F   11 |K 4 ˜) is used. The theorem holds because the members from {K 1 , K 2 , K 3 , K 4 } are assumed to be stochastically self-reliant causes of F 11 . 
         [0049]    This results in: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       a 
                       10 
                     
                     := 
                   
                 
               
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               11 
                             
                              
                             
                                 
                             
                              
                             … 
                              
                             
                                 
                             
                              
                             
                               F 
                               16 
                             
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     = 
                   
                 
               
               
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               F 
                               11 
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       · 
                       … 
                       · 
                       
                         p 
                          
                         
                           ( 
                           
                             
                               F 
                               16 
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                   
                 
               
               
                 
                   
                       
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       11 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               p 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       2 
                                     
                                      
                                     
                                       K 
                                       3 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               p 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       · 
                     
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     
                       · 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         K 
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               p 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     = 
                   
                 
               
               
                 
                   
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     3 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     · 
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     · 
                     
                         
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         K 
                                         2 
                                       
                                       ∼ 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         K 
                                         3 
                                       
                                       ∼ 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         K 
                                         4 
                                       
                                       ∼ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         ; 
                       
                     
                   
                 
               
             
           
         
       
     
         [0050]    Note 3 
         [0051]    In the requirements it was merely demanded that the set {K 1 ′, K 2 ′, K 3 ′, K 4 ′}contains those K′-events that induce two or more F-elements. Accordingly, outside the set {K 1 ′, K 2 ′, K 3 ′, K 4 ′}other causes of F-elements might exist that induce only a single F-element. Then the theorem “factorization in case of hidden causes” is applied, which generates for the exemplary probability p(F 11 |K 1 K 2 K 3 K 4 ) the following: 
         [0000]    
       
         
           
             
               p 
                
               
                 ( 
                 
                   
                     F 
                     11 
                   
                   | 
                   
                     
                       
                         K 
                         _ 
                       
                       1 
                     
                      
                     
                       K 
                       2 
                     
                      
                     
                       K 
                       3 
                     
                      
                     
                       K 
                       4 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 1 
                 - 
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     2 
                   
                 
               
               = 
               
                 1 
                 - 
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               K 
                               2 
                             
                             ∼ 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               K 
                               3 
                             
                             ∼ 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           
                             
                               K 
                               4 
                             
                             ∼ 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             11 
                           
                           | 
                           ∼ 
                         
                         ) 
                       
                     
                     2 
                   
                 
               
             
           
         
       
     
         [0052]    Note 4 
         [0053]    For practical purposes p(F 11 |˜):=0 is set. This is just a preliminary arrangement, which can be revoked at any time for the purposes of improving precision. (In addition, since the hidden cause that might exist creates only one F-element, the conditional stochastic independence of the F-elements is not violated.) 
         [0054]    Note 5 
         [0055]    Also consider that for any F ij  and any K i  the expression p(F ij |K i ˜):=0 always applies if F ij  does not have K i  as a cause. For example, in the previous calculation of a 10  there appears p(F 11 |  K   1 K 2 K 3 K 4 ) which could be zero if F 11  is only caused by K 1  and neither by K 2  nor K 3  nor K 4 . However, this possibility is excluded by the requirement that each F ij  must be caused by at least two K′-elements. 
         [0056]    After linear interpolation of Z 1 , the other members from {a 1k , k:=0, . . . , 7} are determined to: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       
                         a 
                         11 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       11 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               p 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     K 
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       · 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 
                                   p 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           F 
                                           _ 
                                         
                                         16 
                                       
                                       | 
                                       
                                         
                                           
                                             K 
                                             _ 
                                           
                                           1 
                                         
                                          
                                         
                                           K 
                                           2 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           3 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 p 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       2 
                                     
                                      
                                     
                                       K 
                                       3 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       2 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       3 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       · 
                       … 
                       · 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       2 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       3 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         12 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       11 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               p 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       · 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 
                                   p 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           F 
                                           _ 
                                         
                                         16 
                                       
                                       | 
                                       
                                         
                                           
                                             K 
                                             _ 
                                           
                                           1 
                                         
                                          
                                         
                                           K 
                                           2 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           3 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 p 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       2 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       3 
                                     
                                      
                                     
                                       K 
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       2 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       4 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       · 
                       … 
                       · 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       2 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       4 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         13 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 3 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         14 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       2 
                                     
                                      
                                     
                                       K 
                                       3 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       2 
                                     
                                      
                                     
                                       
                                         K 
                                         _ 
                                       
                                       3 
                                     
                                      
                                     
                                       K 
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       · 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         K 
                                         3 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       3 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     11 
                                   
                                   | 
                                   
                                     
                                       K 
                                       4 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       · 
                       … 
                       · 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       3 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       _ 
                                     
                                     16 
                                   
                                   | 
                                   
                                     
                                       K 
                                       4 
                                     
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         15 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         16 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   4 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   4 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                       ; 
                     
                   
                 
               
                
               
                 
 
               
                
               
                 
                   
                     
                       
                         a 
                         17 
                       
                       := 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 3 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       = 
                     
                   
                 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               F 
                               11 
                             
                             | 
                             ∼ 
                           
                           ) 
                         
                       
                       · 
                       … 
                       · 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               ∼ 
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
         [0057]    With the coefficients a 10  to a 17  we obtain for Z 1 :=p(F 11  . . . F 16 |  K   1 K 2 ′K 3 ′K 4 ′): 
         [0000]        Z   1   :=p ( F   11    . . . F   16   |  K     1   ,K   2   ′K   3   ′K   4 ′)= a   10   x   2   x   3   x   4   +a   11   x   2   x   3     x     4   +a   12   x   2   x   3     x     4   +a   13   x   2     x     3     x     4   +a   14     x     2   x   3   x   4   +a   15     x     2   x   3     x     4   +a   16     x     2     x     3   x   4   +a   17     x     2     x     3     x     4 .
 
       Schematic Formation of the Coefficients a ik  and b ik    
       [0058]    For the K i ′-foursome grouping and any x i  we have 
         [0000]    
       
         
           
             
               
                 x 
                 i 
               
               := 
               
                 1 
                 
                   1 
                   + 
                   
                     
                       
                         Z 
                         i 
                       
                       
                         N 
                         i 
                       
                     
                     · 
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               K 
                               _ 
                             
                             i 
                           
                           ) 
                         
                       
                       
                         p 
                          
                         
                           ( 
                           
                             K 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    i:=1, . . . , 4, with 
         [0000]        Z   1   :=p ( F   11    . . . F   16   |  K     1   K   2   ′K   3   ′K   4 ′) and  N   1   :=p ( F   11    . . . F   16   |K   1   K   2   ′K   3   ′K   4 ′),
 
         [0000]        Z   2   :=p ( F   21    . . . F   26   |  K     1   K   2   ′K   3   ′K   4 ′) and  N   3   :=p ( F   21    . . . F   26   |K   2   K   1   ′K   3   ′K   4 ′),
 
         [0000]        Z   3   :=p ( F   31    . . . F   36   |  K     3   K   2   ′K   1   ′K   4 ′) and  N   3   :=p ( F   31    . . . F   36   |K   3   K   2   ′K   1   ′K   4 ′),
 
         [0000]        Z   4   :=p ( F   41    . . . F   46   |  K     4   K   2   ′K   3   ′K   1 ′) and  N   4   :=p ( F   41    . . . F   46   |K   4   K   2   ′K   3   ′K   1 ′).
 
         [0059]    As an example we choose Z 1 :=p(F 11  . . . F 16 |  K   1 K 2 ′K 3 ′K 4 ′) to demonstrate the formation of the coefficients a ik . 
         [0060]    Step 1: 
         [0061]    For any a ik , e.g. for a 14 , the number k situated in the index of a ik  (here it has the value 4) is written as the binary number (100). 
         [0062]    Step 2: 
         [0063]    The binary number (100) is right-aligned projected onto the apostrophized elements in p(F 11  . . . F 16 |  K   1 K 2 ′K 3 ′K 4 ′) that results to 
         [0000]    
       
                 
         
             
             
         
       
     
         [0000]    whereby the apostrophes are then omitted, and the digits “1” of the binary numbers indicate the negations to be executed; in the example it leads to 
         [0000]        a   14   :=p ( F   11    . . . F   16   |  K     1     K     2   K   3   K   4 ). 
         [0064]    Step 3: 
         [0065]    It follows a factorization with respect to the F-elements: 
         [0000]        p ( F   11    . . . F   16   |  K     1     K     2   K   3   K   4 )= p ( F   11   |  K     1     K     2   K   3   K   4 )·. . . · p ( F   16   |  K     1     K     2   K   3   K   4 ).
 
         [0066]    This is followed by a factorization with respect to the K-elements. For this purpose, a simple pattern can be used, for example 
         [0000]        p ( F   11   |  K     1     K     2   K   3   K   4 ):=1− p (   F     11   |  K     1 ˜)· p (   F     11   |  K     2 ˜)· p (   F     11   |  K     3 ˜)· p (   F     11   |  K     4 ˜)
 
         [0000]    which due to p(F ij |˜]:=0 is shortened to 
         [0000]        p ( F   11   |  K     1     K     2   K   3   K   4 ):=1− p (   F     11   |K   3 ˜)· p (   F     11   |K   4 ˜).
 
         [0067]    Step 4: 
         [0068]    The coefficient a 14  belongs to a product of unknowns. The individual elements of this product have the same negations and indices as those obtained in Step 2, i.e. the projection is continued directly to 
         [0000]    
       
                 
         
             
             
         
       
     
         [0069]    The determined (  x   2 x 3 x 4 ) is the product of unknowns associated with the coefficient a 14 . 
         [0070]    Result: 
         [0000]        a   14     x     2   x   3   x   4 =[1− p (   F     11   |K   3  . . . )· p (   F     11   |K   4  . . . )]· . . . ·[1− p (   F     16   |K   3  . . . )· p (   F     16   |K   4  . . . )]·   x     2   x   3   x   4 .
 
         [0000]    Completely in line with this approach, i.e. after linear interpolation of N i  or the application of the above scheme upon N 1 , the coefficients b 1k , k:=0, . . . , 7 are formed, thus resulting in the following: 
         [0000]    
       
         
           
             
               N 
               1 
             
             := 
             
               
                 p 
                  
                 
                   ( 
                   
                     
                       
                         F 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         F 
                         16 
                       
                     
                     | 
                     
                       
                         K 
                         1 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     b 
                     10 
                   
                    
                   
                     x 
                     2 
                   
                    
                   
                     x 
                     3 
                   
                    
                   
                     x 
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     11 
                   
                    
                   
                     x 
                     2 
                   
                    
                   
                     x 
                     3 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     12 
                   
                    
                   
                     x 
                     2 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     3 
                   
                    
                   
                     x 
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     13 
                   
                    
                   
                     x 
                     2 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     3 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     14 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     2 
                   
                    
                   
                     x 
                     3 
                   
                    
                   
                     x 
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     15 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     2 
                   
                    
                   
                     x 
                     3 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     16 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     2 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     3 
                   
                    
                   
                     x 
                     4 
                   
                 
                 + 
                 
                   
                     b 
                     17 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     2 
                   
                    
                   
                     
                       x 
                       _ 
                     
                     3 
                   
                    
                   
                     
                       
                         x 
                         _ 
                       
                       4 
                     
                     . 
                     
                       
 
                     
                      
                     
                       
                         
                           
                             
                               b 
                               10 
                             
                             := 
                           
                         
                       
                       
                         
                           
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       F 
                                       11 
                                     
                                      
                                     
                                         
                                     
                                      
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     
                                       F 
                                       16 
                                     
                                   
                                   | 
                                   
                                     
                                       K 
                                       1 
                                     
                                      
                                     
                                       K 
                                       2 
                                     
                                      
                                     
                                       K 
                                       3 
                                     
                                      
                                     
                                       K 
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                             = 
                           
                         
                       
                       
                         
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       F 
                                       11 
                                     
                                     | 
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         K 
                                         3 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               … 
                               · 
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       F 
                                       16 
                                     
                                     | 
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         K 
                                         2 
                                       
                                        
                                       
                                         K 
                                         3 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             = 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             K 
                                             1 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             2 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             3 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             
                                               K 
                                               _ 
                                             
                                             1 
                                           
                                            
                                           
                                             K 
                                             2 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             3 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   p 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           F 
                                           _ 
                                         
                                         11 
                                       
                                       | 
                                       
                                         
                                           
                                             K 
                                             _ 
                                           
                                           1 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           2 
                                         
                                          
                                         
                                           K 
                                           3 
                                         
                                          
                                         
                                           
                                             K 
                                             _ 
                                           
                                           4 
                                         
                                       
                                     
                                     ) 
                                   
                                   · 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       11 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             · 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             · 
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               K 
                                               1 
                                             
                                              
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               2 
                                             
                                              
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               3 
                                             
                                              
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               4 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               1 
                                             
                                              
                                             
                                               K 
                                               2 
                                             
                                              
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               3 
                                             
                                              
                                             
                                               
                                                 K 
                                                 _ 
                                               
                                               4 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     p 
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           16 
                                         
                                         | 
                                         
                                           
                                             
                                               K 
                                               _ 
                                             
                                             1 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             2 
                                           
                                            
                                           
                                             K 
                                             3 
                                           
                                            
                                           
                                             
                                               K 
                                               _ 
                                             
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                     · 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 p 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         _ 
                                       
                                       16 
                                     
                                     | 
                                     
                                       
                                         
                                           K 
                                           _ 
                                         
                                         1 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           K 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         K 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             = 
                           
                         
                       
                       
                         
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             K 
                                             1 
                                           
                                           ∼ 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             K 
                                             2 
                                           
                                           ∼ 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             K 
                                             3 
                                           
                                           ∼ 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     p 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             F 
                                             _ 
                                           
                                           11 
                                         
                                         | 
                                         
                                           
                                             K 
                                             4 
                                           
                                           ∼ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                             · 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               · 
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               K 
                                               1 
                                             
                                             ∼ 
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               K 
                                               2 
                                             
                                             ∼ 
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               K 
                                               3 
                                             
                                             ∼ 
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     
                                       p 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               F 
                                               _ 
                                             
                                             16 
                                           
                                           | 
                                           
                                             
                                               K 
                                               4 
                                             
                                             ∼ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ; 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     11 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             K 
                             2 
                           
                            
                           
                             K 
                             3 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           p 
                         
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 _ 
                               
                               11 
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 K 
                                 3 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   · 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             16 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     3 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     12 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             K 
                             2 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             3 
                           
                            
                           
                             K 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           p 
                         
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 _ 
                               
                               11 
                             
                             | 
                             
                               
                                 
                                   K 
                                   _ 
                                 
                                 1 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 2 
                               
                                
                               
                                 
                                   K 
                                   _ 
                                 
                                 3 
                               
                                
                               
                                 K 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   · 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             16 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     13 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             K 
                             2 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             3 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   … 
                   · 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   2 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     14 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             2 
                           
                            
                           
                             K 
                             3 
                           
                            
                           
                             K 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   · 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               _ 
                             
                             16 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   4 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     3 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     15 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             2 
                           
                            
                           
                             K 
                             3 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     K 
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   … 
                   · 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     16 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             2 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             3 
                           
                            
                           
                             K 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     K 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     · 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       K 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   4 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   … 
                   · 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   4 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     b 
                     17 
                   
                   := 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           
                             F 
                             11 
                           
                            
                           
                               
                           
                            
                           … 
                            
                           
                               
                           
                            
                           
                             F 
                             16 
                           
                         
                         | 
                         
                           
                             K 
                             1 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             2 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             3 
                           
                            
                           
                             
                               K 
                               _ 
                             
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     … 
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               2 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               3 
                             
                              
                             
                               
                                 K 
                                 _ 
                               
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                 
               
             
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           F 
                           11 
                         
                         | 
                         
                           
                             K 
                             1 
                           
                           ∼ 
                         
                       
                       ) 
                     
                   
                   · 
                   … 
                   · 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               K 
                               1 
                             
                             ∼ 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0071]    Note 6 
         [0072]    In the transformation set out below it can be seen how the a-priori probabilities, e.g. p(K 1 ), get an upgrade. 
         [0000]    
       
         
           
             
               p 
                
               
                 ( 
                 
                   
                     K 
                     1 
                   
                   | 
                   
                     
                       F 
                       11 
                     
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                       F 
                       16 
                     
                      
                     
                       K 
                       2 
                       ′ 
                     
                      
                     
                       K 
                       3 
                       ′ 
                     
                      
                     
                       K 
                       4 
                       ′ 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   p 
                    
                   
                     ( 
                     
                       
                         F 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         F 
                         16 
                       
                        
                       
                         K 
                         1 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
                 
                   p 
                    
                   
                     ( 
                     
                       
                         F 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         F 
                         16 
                       
                        
                       
                         K 
                         2 
                         ′ 
                       
                        
                       
                         K 
                         3 
                         ′ 
                       
                        
                       
                         K 
                         4 
                         ′ 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               11 
                             
                              
                             
                                 
                             
                              
                             … 
                              
                             
                                 
                             
                              
                             
                               F 
                               16 
                             
                           
                           | 
                           
                             
                               K 
                               1 
                             
                              
                             
                               K 
                               2 
                               ′ 
                             
                              
                             
                               K 
                               3 
                               ′ 
                             
                              
                             
                               K 
                               4 
                               ′ 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             K 
                             1 
                           
                            
                           
                             K 
                             2 
                             ′ 
                           
                            
                           
                             K 
                             3 
                             ′ 
                           
                            
                           
                             K 
                             4 
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                   
                     p 
                      
                     
                       ( 
                       
                         
                           F 
                           11 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           16 
                         
                          
                         
                           K 
                           2 
                           ′ 
                         
                          
                         
                           K 
                           3 
                           ′ 
                         
                          
                         
                           K 
                           4 
                           ′ 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 K 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 16 
                               
                             
                             | 
                             
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                     
                     · 
                     
                       p 
                        
                       
                         ( 
                         
                           
                             K 
                             1 
                           
                           | 
                           
                             
                               K 
                               2 
                               ′ 
                             
                              
                             
                               K 
                               3 
                               ′ 
                             
                              
                             
                               K 
                               4 
                               ′ 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                       
                     
                       
                         [ 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     11 
                                   
                                    
                                   
                                       
                                   
                                    
                                   … 
                                    
                                   
                                       
                                   
                                    
                                   
                                     F 
                                     16 
                                   
                                 
                                 | 
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     3 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     4 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     11 
                                   
                                    
                                   
                                       
                                   
                                    
                                   … 
                                    
                                   
                                       
                                   
                                    
                                   
                                     F 
                                     16 
                                   
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     3 
                                     ′ 
                                   
                                    
                                   
                                     K 
                                     4 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       · 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               K 
                               1 
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
         [0073]    Now it can be seen immediately that the a-priori probability p(K i ) gets an upgrade by the appreciation factor 
         [0000]    
       
         
           
             
               AF 
                
               
                 ( 
                 1 
                 ) 
               
             
             := 
             
               
                 
                   x 
                   1 
                 
                 
                   p 
                    
                   
                     ( 
                     
                       K 
                       1 
                     
                     ) 
                   
                 
               
               = 
               
                   
                 
                   
                     [ 
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 1 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 6 
                               
                             
                             | 
                             
                               
                                 K 
                                 1 
                               
                                
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                       
                         p 
                          
                         
                           ( 
                           
                             
                               
                                 F 
                                 1 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 6 
                               
                             
                             | 
                             
                               
                                 K 
                                 2 
                                 ′ 
                               
                                
                               
                                 K 
                                 3 
                                 ′ 
                               
                                
                               
                                 K 
                                 4 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                   . 
                 
               
             
           
         
       
     
         [0074]    Note 7 
         [0075]    Generally, the appreciation factor AF(i) shows by what factor the a-priori probability p(K i ) changes, and it thus measures the influence of the evaluation environment W(K i ′) upon the probability of existence of K i ′, stating a veritable numerical value. 
         [0076]    Note 8 
         [0077]    The term p(F 11  . . . F 16 |K 2 ′K 3 ′K 4 ′) in the denominator of the equation above can—after linear interpolation—not be factorized with respect to the F-elements, because the condition does not contain all causes of the F-elements; an expansion is therefore required as it was performed for x 1  at the very beginning. 
         [0078]    Initially, equiprobability is assumed for all K′-elements, i.e. the a-priori probabilities p(K i ), i:=1, . . . , 4 are set to p(K i ): =0.25. After performing the calculations, i.e. when the a-posteriori probabilities x i , i:=1, . . . , 4 are available, the appreciation factors 
         [0000]    
       
         
           
             
               AF 
                
               
                 ( 
                 i 
                 ) 
               
             
             = 
             
               
                 x 
                 1 
               
               
                 p 
                  
                 
                   ( 
                   
                     K 
                     i 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    can be formed, whereby AF(i)&lt;1 represents a downgrading and AF(i)&gt;1 an upgrading. The highest appreciation factor indicates that diagnosis for which the probability has risen most clearly, and which therefore becomes the first and foremost to be considered as the cause for the symptoms in question. 
         [0079]    In order to determine the true probabilities of existence for the diagnoses K i ′, the “true” numerical values for the a-priori probabilities are required. For this purpose, a basic quantity is defined, e.g. the number of emergency patients who were treated within a certain period of time due to heart problems by an emergency doctor. For a group of K′-elements, such as for the foursome grouping {K 1 ′, K 2 ′, K 3 ′, K 4 ′} concerning cardiac diagnoses, the “true” a-priori probabilities p(K i ), i:=1, . . . , 4 can at least be approximated by counting the relative frequencies h(K i ). The diagnoses calculated by the method presented are then valid when the p(K i ) values obtained this way are used only for the case of the defined basic set. 
         [0080]    In anticipation of future tasks, a clear improvement of the method can be achieved by using the events from the next higher level, i.e. the causes of the causative K′-elements. For example, the a-priori probability p(K i ) is replaced by p (K 1 |U 11  U 12  U 13 ) where the arbitrarily established set {U 11 , U 12 , U 13 } includes the causes of K i . The investigation for possible causes of the K′-elements allows improved individualization of cases treated. 
         [0081]    The desired calculation of the unknown x 1  is executed by the use of c 1 : =p(K i ) as follows: 
         [0000]    
       
         
           
             
               x 
               1 
             
             := 
             
               
                 ( 
                 
                   
                     K 
                     1 
                   
                   | 
                   
                     
                       F 
                       11 
                     
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                       F 
                       16 
                     
                      
                     
                       K 
                       2 
                       ′ 
                     
                      
                     
                       K 
                       3 
                       ′ 
                     
                      
                     
                       K 
                       4 
                       ′ 
                     
                   
                 
                 ) 
               
               = 
               
                 
                   1 
                   
                     1 
                     + 
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   11 
                                 
                                  
                                 
                                     
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   F 
                                   16 
                                 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                   ′ 
                                 
                                  
                                 
                                   K 
                                   3 
                                   ′ 
                                 
                                  
                                 
                                   K 
                                   4 
                                   ′ 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 
                                   F 
                                   11 
                                 
                                  
                                 
                                     
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   F 
                                   16 
                                 
                               
                               | 
                               
                                 
                                   K 
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                   ′ 
                                 
                                  
                                 
                                   K 
                                   3 
                                   ′ 
                                 
                                  
                                 
                                   K 
                                   4 
                                   ′ 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           p 
                            
                           
                             ( 
                             
                               K 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       1 
                       
                         1 
                         + 
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         a 
                                         10 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         11 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         12 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         13 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           a 
                                           14 
                                         
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         15 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         16 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         17 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         b 
                                         10 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         11 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         12 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         13 
                                       
                                        
                                       
                                         x 
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           b 
                                           14 
                                         
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         15 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         x 
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         16 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         x 
                                         4 
                                       
                                     
                                     + 
                                     
                                       
                                         b 
                                         17 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         3 
                                       
                                        
                                       
                                         
                                           x 
                                           _ 
                                         
                                         4 
                                       
                                     
                                   
                                 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   c 
                                   _ 
                                 
                                 1 
                               
                               
                                 c 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                     . 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       AF 
                        
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                   := 
                   
                     
                       
                         x 
                         1 
                       
                       
                         c 
                         1 
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0082]    The equations for calculating the unknowns x 2 , x 3  and x 4  are set up in exactly the same way. This gives a system of equations in four unknowns which is solved by means of a commercially available calculation program. 
         [0000]    
       
         
           
             
               
                 eq 
                  
                 
                     
                 
                  
                 1 
               
               := 
               
                 
                   x 
                   1 
                 
                 = 
                 
                   1 
                   
                     1 
                     + 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     10 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     11 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     12 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     13 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     a 
                                     14 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     15 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     16 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     17 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     b 
                                     10 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     11 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     12 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     13 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     b 
                                     14 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     15 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     16 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     17 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               c 
                               _ 
                             
                             1 
                           
                           
                             c 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               eq 
                
               
                   
               
                
               2 
             
             := 
             
               
                 x 
                 2 
               
               = 
               
                 1 
                 
                   1 
                   + 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   20 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   21 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   22 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   23 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   a 
                                   24 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   25 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   26 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   27 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   b 
                                   20 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   21 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   22 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   23 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   24 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   25 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   26 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   27 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   4 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             c 
                             _ 
                           
                           2 
                         
                         
                           c 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 eq 
                  
                 
                     
                 
                  
                 3 
               
               := 
               
                 
                   x 
                   3 
                 
                 = 
                 
                   1 
                   
                     1 
                     + 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     30 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     31 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     32 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     33 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     a 
                                     34 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     35 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     36 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     37 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     b 
                                     30 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     31 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     32 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     33 
                                   
                                    
                                   
                                     x 
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     b 
                                     34 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     35 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     36 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     b 
                                     37 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     
                                       x 
                                       _ 
                                     
                                     4 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               c 
                               _ 
                             
                             3 
                           
                           
                             c 
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               eq 
                
               
                   
               
                
               4 
             
             := 
             
               
                 x 
                 4 
               
               = 
               
                 1 
                 
                   1 
                   + 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   40 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   41 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   42 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   43 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   a 
                                   44 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   45 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   46 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   47 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   b 
                                   40 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   41 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   42 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   43 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   44 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   45 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   x 
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   46 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   b 
                                   47 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   3 
                                 
                                  
                                 
                                   
                                     x 
                                     _ 
                                   
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             c 
                             _ 
                           
                           4 
                         
                         
                           c 
                           4 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   AF 
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
                 := 
                 
                   
                     x 
                     1 
                   
                   
                     c 
                     1 
                   
                 
               
               ; 
               
                 
                   AF 
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
                 := 
                 
                   
                     x 
                     2 
                   
                   
                     c 
                     2 
                   
                 
               
               ; 
               
                 
                   AF 
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
                 := 
                 
                   
                     x 
                     3 
                   
                   
                     c 
                     3 
                   
                 
               
               ; 
               
                 
                   AF 
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
                 := 
                 
                   
                     
                       x 
                       4 
                     
                     
                       c 
                       4 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0083]    As an example, a complete and without modifications directly workable program for four unknowns is given below. 
         [0084]    As an indicator of the presence or absence of an event from the set {F ij } we introduce the factor f ij , with f ij  from {0,1}. 
         [0085]    The coefficient a 10  serves as an example to illustrate the changes in a ik  and b ik . As it has already been shown in the preceding description, a 10  was found to be: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     := 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         p 
                          
                         
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     ; 
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     := 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         p 
                          
                         
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     ; 
                   
                 
               
               
                 
                   
                     
                       
                         a 
                         10 
                       
                       := 
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         … 
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 16 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ; 
                   
                 
               
             
             . 
           
         
       
     
         [0086]    In order to determine the possibility that elements from {F 11 , . . . , F 16 } are not present as symptoms, the factors f ij  are formed, so that from {F 11 , . . . , F 16 } the present or absent events can be marked using f ij  as follows: 
         [0087]    If F ij  is present as a symptom, then f ij : =1. 
         [0088]    If F ij  is not present as a symptom, then f ij : =0. 
         [0089]    The indicators f ij  are the a ik  and b ik  supplements, changing the example a 10  to: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             11 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     := 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         p 
                          
                         
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 11 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   11 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     ; 
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     
                       p 
                        
                       
                         ( 
                         
                           
                             F 
                             16 
                           
                           | 
                           
                             
                               
                                 K 
                                 _ 
                               
                               1 
                             
                              
                             
                               K 
                               2 
                             
                              
                             
                               K 
                               3 
                             
                              
                             
                               K 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     := 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     2 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                           · 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         p 
                          
                         
                           
                             ( 
                             
                               
                                 
                                   F 
                                   _ 
                                 
                                 16 
                               
                               | 
                               
                                 
                                   K 
                                   3 
                                 
                                 ∼ 
                               
                             
                             ) 
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     F 
                                     _ 
                                   
                                   16 
                                 
                                 | 
                                 
                                   
                                     K 
                                     4 
                                   
                                   ∼ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     ; 
                   
                 
               
               
                 
                   
                     
                       a 
                       10 
                     
                     := 
                   
                 
               
               
                 
                   
                     [ 
                     
                       
                         
                           f 
                           11 
                         
                         · 
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 F 
                                 11 
                               
                               | 
                               
                                 
                                   
                                     K 
                                     _ 
                                   
                                   1 
                                 
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                                 
                                   K 
                                   3 
                                 
                                  
                                 
                                   K 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               f 
                               11 
                             
                           
                           ) 
                         
                         · 
                         
                           ( 
                           
                             1 
                             - 
                             
                               p 
                                
                               
                                 ( 
                                 
                                   
                                     F 
                                     11 
                                   
                                   | 
                                   
                                     
                                       
                                         K 
                                         _ 
                                       
                                       1 
                                     
                                      
                                     K 
                                      
                                     
                                         
                                     
                                      
                                     
                                       K 
                                       3 
                                     
                                      
                                     
                                       K 
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         · 
                       
                     
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     · 
                     
                       [ 
                       
                         
                           
                             f 
                             16 
                           
                           · 
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   F 
                                   16 
                                 
                                 | 
                                 
                                   
                                     
                                       K 
                                       _ 
                                     
                                     1 
                                   
                                    
                                   
                                     K 
                                     2 
                                   
                                    
                                   
                                     K 
                                     3 
                                   
                                    
                                   
                                     K 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 f 
                                 16 
                               
                             
                             ) 
                           
                           · 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   p 
                                    
                                   
                                     ( 
                                     
                                       
                                         F 
                                         16 
                                       
                                       | 
                                       
                                         
                                           
                                             K 
                                             _ 
                                           
                                           1 
                                         
                                          
                                         
                                           K 
                                           2 
                                         
                                          
                                         
                                           K 
                                           3 
                                         
                                          
                                         
                                           K 
                                           4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0090]    These factors f ij  are to be attached to all a ik  and b ik . 
         [0091]    It follows the announced complete and fully workable example which was built on the basis of Table 1, 2 and 3. This in great detail presented example may serve as a defining example.