:: A Projective Closure and Projective Horizon of an Affine Space
::  by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski

environ

 vocabularies DIRAF, SUBSET_1, STRUCT_0, AFF_4, INCSP_1, AFF_1, ANALOAF,
      RELAT_1, TARSKI, PARSP_1, XBOOLE_0, ARYTM_3, SETFAM_1, ZFMISC_1, EQREL_1,
      RELAT_2, ANPROJ_1, INCPROJ, MCART_1, FDIFF_1, ANPROJ_2, AFF_2, VECTSP_1,
      AFPROJ;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, DOMAIN_1, EQREL_1, RELSET_1,
      RELAT_1, RELAT_2, XTUPLE_0, MCART_1, STRUCT_0, ANALOAF, DIRAF, AFF_1,
      AFF_4, AFF_2, PAPDESAF, INCSP_1, INCPROJ;
 constructors DOMAIN_1, EQREL_1, AFF_1, AFF_2, TRANSLAC, INCPROJ, AFF_4,
      RELSET_1, XTUPLE_0;
 registrations XBOOLE_0, SUBSET_1, RELSET_1, STRUCT_0, INCPROJ, XTUPLE_0;
 requirements SUBSET, BOOLE;
 definitions TARSKI;
 equalities TARSKI;
 expansions TARSKI;
 theorems RELAT_1, RELAT_2, TARSKI, ZFMISC_1, EQREL_1, AFF_1, AFF_4, INCPROJ,
      PAPDESAF, AFF_2, DIRAF, INCSP_1, XBOOLE_0, PARTFUN1, ORDERS_1, XTUPLE_0;
 schemes RELSET_1;

begin

reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;

:: The aim of this article is to formalize the well known construction of
:: the projective closure of an affine space. We begin with some evident
:: properties of planes in affine planes.

theorem Th1:
  AS is AffinPlane & X=the carrier of AS implies X is being_plane
proof
  assume that
A1: AS is AffinPlane and
A2: X=the carrier of AS;
  consider a,b,c being Element of AS such that
A3: not LIN a,b,c by AFF_1:12;
  set P=Line(a,b),K=Line(a,c);
A4: b in P by AFF_1:15;
A5: c in K by AFF_1:15;
  a<>b by A3,AFF_1:7;
  then
A6: P is being_line by AFF_1:def 3;
  set Y=Plane(K,P);
A7: a in P by AFF_1:15;
  a<>c by A3,AFF_1:7;
  then
A8: K is being_line by AFF_1:def 3;
A9: a in K by AFF_1:15;
A10: not K // P
  proof
    assume K // P;
    then c in P by A7,A9,A5,AFF_1:45;
    hence contradiction by A3,A7,A4,A6,AFF_1:21;
  end;
  now
    let x be object;
    assume x in X;
    then reconsider a=x as Element of AS;
    set K9=a*K;
A11: K9 is being_line by A8,AFF_4:27;
A12: K // K9 by A8,AFF_4:def 3;
    then not K9 // P by A10,AFF_1:44;
    then consider b being Element of AS such that
A13: b in K9 and
A14: b in P by A1,A6,A11,AFF_1:58;
    a in K9 by A8,AFF_4:def 3;
    then a,b // K by A12,A13,AFF_1:40;
    then a in {zz: ex b being Element of AS st zz,b // K & b in P} by A14;
    hence x in Y by AFF_4:def 1;
  end;
  then
A15: X c= Y;
  Y is being_plane by A6,A8,A10,AFF_4:def 2;
  hence thesis by A2,A15,XBOOLE_0:def 10;
end;

theorem Th2:
  AS is AffinPlane & X is being_plane implies X = the carrier of AS
proof
  assume that
A1: AS is AffinPlane and
A2: X is being_plane;
  the carrier of AS c= the carrier of AS;
  then reconsider Z=the carrier of AS as Subset of AS;
  Z is being_plane by A1,Th1;
  hence thesis by A2,AFF_4:33;
end;

theorem Th3:
  AS is AffinPlane & X is being_plane & Y is being_plane implies X= Y
proof
  assume that
A1: AS is AffinPlane and
A2: X is being_plane and
A3: Y is being_plane;
  X=the carrier of AS by A1,A2,Th2;
  hence thesis by A1,A3,Th2;
end;

theorem
  X=the carrier of AS & X is being_plane implies AS is AffinPlane
proof
  assume that
A1: X=the carrier of AS and
A2: X is being_plane;
  assume AS is not AffinPlane;
  then ex zz st not zz in X by A2,AFF_4:48;
  hence contradiction by A1;
end;

theorem Th5:
  not A // K & A '||' X & A '||' Y & K '||' X & K '||' Y & A is
  being_line & K is being_line & X is being_plane & Y is being_plane implies X
  '||' Y
proof
  assume that
A1: not A // K and
A2: A '||' X and
A3: A '||' Y and
A4: K '||' X and
A5: K '||' Y and
A6: A is being_line and
A7: K is being_line and
A8: X is being_plane and
A9: Y is being_plane;
  set y = the Element of Y;
  set x = the Element of X;
A10: Y <> {} by A9,AFF_4:59;
A11: X <> {} by A8,AFF_4:59;
  then reconsider a=x,b=y as Element of AS by A10,TARSKI:def 3;
A12: K // a*K by A7,AFF_4:def 3;
A13: A // a*A by A6,AFF_4:def 3;
A14: not a*A // a*K
  proof
    assume not thesis;
    then a*A // K by A12,AFF_1:44;
    hence contradiction by A1,A13,AFF_1:44;
  end;
  a*K c= a+X by A4,A7,A8,AFF_4:68;
  then
A15: a*K c= X by A8,A11,AFF_4:66;
  K // b*K by A7,AFF_4:def 3;
  then
A16: a*K // b*K by A12,AFF_1:44;
  b*A c= b+Y by A3,A6,A9,AFF_4:68;
  then
A17: b*A c= Y by A9,A10,AFF_4:66;
  A // b*A by A6,AFF_4:def 3;
  then
A18: a*A // b*A by A13,AFF_1:44;
  b*K c= b+Y by A5,A7,A9,AFF_4:68;
  then
A19: b*K c= Y by A9,A10,AFF_4:66;
  a*A c= a+X by A2,A6,A8,AFF_4:68;
  then a*A c= X by A8,A11,AFF_4:66;
  hence thesis by A8,A9,A15,A17,A19,A14,A18,A16,AFF_4:55;
end;

theorem
  X is being_plane & A '||' X & X '||' Y implies A '||' Y by AFF_4:59,60;

:: Next we distinguish two sets, one consisting of the lines, the second
:: consisting of the planes of a given affine space and we consider two
:: equivalence relations defined on each of these sets; theses relations
:: are in fact the relation of parallelity restricted to suitable area.
:: Their equivalence classes are called directions (of lines and planes,
:: respectively); they are intended to be considered as new objects which
:: extend the given affine space to a projective space.

definition
  let AS;
  func AfLines(AS) -> Subset-Family of AS equals
  {A: A is being_line};
  coherence
  proof
    set X={A: A is being_line};
    X c= bool the carrier of AS
    proof
      let x be object;
      assume x in X;
      then ex A st x=A & A is being_line;
      hence thesis;
    end;
    hence thesis;
  end;
end;

definition
  let AS;
  func AfPlanes(AS) -> Subset-Family of AS equals
  {A: A is being_plane};
  coherence
  proof
    set X={A: A is being_plane};
    X c= bool the carrier of AS
    proof
      let x be object;
      assume x in X;
      then ex A st x=A & A is being_plane;
      hence thesis;
    end;
    hence thesis;
  end;
end;

theorem
  for x holds (x in AfLines(AS) iff ex X st x=X & X is being_line);

theorem
  for x holds (x in AfPlanes(AS) iff ex X st x=X & X is being_plane);

definition
  let AS;
  func LinesParallelity(AS) -> Equivalence_Relation of AfLines(AS) equals
  {[K,
  M]: K is being_line & M is being_line & K '||' M};
  coherence
  proof
    set AFL=AfLines(AS),AFL2=[:AfLines(AS),AfLines(AS):];
    set R1={[X,Y]: X is being_line & Y is being_line & X '||' Y};
    now
      let x be object;
      assume x in R1;
      then consider X,Y such that
A1:   x=[X,Y] and
A2:   X is being_line and
A3:   Y is being_line and
      X '||' Y;
A4:   Y in AFL by A3;
      X in AFL by A2;
      hence x in AFL2 by A1,A4,ZFMISC_1:def 2;
    end;
    then reconsider R2=R1 as Relation of AFL,AFL by TARSKI:def 3;
    now
      let x be object;
      assume x in AFL;
      then consider X such that
A5:   x=X and
A6:   X is being_line;
      X // X by A6,AFF_1:41;
      then X '||' X by A6,AFF_4:40;
      hence [x,x] in R2 by A5,A6;
    end;
    then
A7: R2 is_reflexive_in AFL by RELAT_2:def 1;
    then
A8: field R2 = AFL by ORDERS_1:13;
A9: X is being_line & Y is being_line implies ([X,Y] in R1 iff X '||' Y)
    proof
      assume that
A10:  X is being_line and
A11:  Y is being_line;
      now
        assume [X,Y] in R1;
        then consider X9,Y9 such that
A12:    [X,Y]=[X9,Y9] and
        X9 is being_line and
        Y9 is being_line and
A13:    X9 '||' Y9;
        X=X9 by A12,XTUPLE_0:1;
        hence X '||' Y by A12,A13,XTUPLE_0:1;
      end;
      hence thesis by A10,A11;
    end;
    now
      let x,y,z be object;
      assume that
A14:  x in AFL and
A15:  y in AFL and
A16:  z in AFL and
A17:  [x,y] in R2 and
A18:  [y,z] in R2;
      consider Y such that
A19:  y=Y and
A20:  Y is being_line by A15;
      consider Z such that
A21:  z=Z and
A22:  Z is being_line by A16;
      Y '||' Z by A9,A18,A19,A20,A21,A22;
      then
A23:  Y // Z by A20,A22,AFF_4:40;
      consider X such that
A24:  x=X and
A25:  X is being_line by A14;
      X '||' Y by A9,A17,A24,A25,A19,A20;
      then X // Y by A25,A20,AFF_4:40;
      then X // Z by A23,AFF_1:44;
      then X '||' Z by A25,A22,AFF_4:40;
      hence [x,z] in R2 by A24,A25,A21,A22;
    end;
    then
A26: R2 is_transitive_in AFL by RELAT_2:def 8;
    now
      let x,y be object;
      assume that
A27:  x in AFL and
A28:  y in AFL and
A29:  [x,y] in R2;
      consider X such that
A30:  x=X and
A31:  X is being_line by A27;
      consider Y such that
A32:  y=Y and
A33:  Y is being_line by A28;
      X '||' Y by A9,A29,A30,A31,A32,A33;
      then X // Y by A31,A33,AFF_4:40;
      then Y '||' X by A31,A33,AFF_4:40;
      hence [y,x] in R2 by A30,A31,A32,A33;
    end;
    then
A34: R2 is_symmetric_in AFL by RELAT_2:def 3;
    dom R2 = AFL by A7,ORDERS_1:13;
    hence thesis by A8,A34,A26,PARTFUN1:def 2,RELAT_2:def 11,def 16;
  end;
end;

definition
  let AS;
  func PlanesParallelity(AS) -> Equivalence_Relation of AfPlanes(AS) equals
  {[
  X,Y]: X is being_plane & Y is being_plane & X '||' Y};
  coherence
  proof
    set AFP=AfPlanes(AS),AFP2=[:AfPlanes(AS),AfPlanes(AS):];
    set R1={[X,Y]: X is being_plane & Y is being_plane & X '||' Y};
    now
      let x be object;
      assume x in R1;
      then consider X,Y such that
A1:   x=[X,Y] and
A2:   X is being_plane and
A3:   Y is being_plane and
      X '||' Y;
A4:   Y in AFP by A3;
      X in AFP by A2;
      hence x in AFP2 by A1,A4,ZFMISC_1:def 2;
    end;
    then reconsider R2=R1 as Relation of AFP,AFP by TARSKI:def 3;
    now
      let x be object;
      assume x in AFP;
      then consider X such that
A5:   x=X and
A6:   X is being_plane;
      X '||' X by A6,AFF_4:57;
      hence [x,x] in R2 by A5,A6;
    end;
    then
A7: R2 is_reflexive_in AFP by RELAT_2:def 1;
    then
A8: field R2 = AFP by ORDERS_1:13;
A9: X is being_plane & Y is being_plane implies ([X,Y] in R1 iff X '||' Y)
    proof
      assume that
A10:  X is being_plane and
A11:  Y is being_plane;
      now
        assume [X,Y] in R1;
        then consider X9,Y9 such that
A12:    [X,Y]=[X9,Y9] and
        X9 is being_plane and
        Y9 is being_plane and
A13:    X9 '||' Y9;
        X=X9 by A12,XTUPLE_0:1;
        hence X '||' Y by A12,A13,XTUPLE_0:1;
      end;
      hence thesis by A10,A11;
    end;
    now
      let x,y,z be object;
      assume that
A14:  x in AFP and
A15:  y in AFP and
A16:  z in AFP and
A17:  [x,y] in R2 and
A18:  [y,z] in R2;
      consider X such that
A19:  x=X and
A20:  X is being_plane by A14;
      consider Y such that
A21:  y=Y and
A22:  Y is being_plane by A15;
      consider Z such that
A23:  z=Z and
A24:  Z is being_plane by A16;
A25:  Y '||' Z by A9,A18,A21,A22,A23,A24;
      X '||' Y by A9,A17,A19,A20,A21,A22;
      then X '||' Z by A20,A22,A24,A25,AFF_4:61;
      hence [x,z] in R2 by A19,A20,A23,A24;
    end;
    then
A26: R2 is_transitive_in AFP by RELAT_2:def 8;
    now
      let x,y be object;
      assume that
A27:  x in AFP and
A28:  y in AFP and
A29:  [x,y] in R2;
      consider X such that
A30:  x=X and
A31:  X is being_plane by A27;
      consider Y such that
A32:  y=Y and
A33:  Y is being_plane by A28;
      X '||' Y by A9,A29,A30,A31,A32,A33;
      then Y '||' X by A31,A33,AFF_4:58;
      hence [y,x] in R2 by A30,A31,A32,A33;
    end;
    then
A34: R2 is_symmetric_in AFP by RELAT_2:def 3;
    dom R2 = AFP by A7,ORDERS_1:13;
    hence thesis by A8,A34,A26,PARTFUN1:def 2,RELAT_2:def 11,def 16;
  end;
end;

definition
  let AS,X;
  func LDir(X) -> Subset of AfLines(AS) equals
  Class(LinesParallelity(AS),X);
  correctness;
end;

definition
  let AS,X;
  func PDir(X) -> Subset of AfPlanes(AS) equals
  Class(PlanesParallelity(AS),X);
  correctness;
end;

theorem Th9:
  X is being_line implies for x holds x in LDir(X) iff ex Y st x=Y
  & Y is being_line & X '||' Y
proof
  assume
A1: X is being_line;
  let x;
A2: now
    assume x in LDir(X);
    then [x,X] in LinesParallelity(AS) by EQREL_1:19;
    then consider K,M such that
A3: [x,X]=[K,M] and
A4: K is being_line and
A5: M is being_line and
A6: K '||' M;
    take Y=K;
A7: X=M by A3,XTUPLE_0:1;
    K // M by A4,A5,A6,AFF_4:40;
    hence x=Y & Y is being_line & X '||' Y by A3,A4,A5,A7,AFF_4:40,XTUPLE_0:1;
  end;
  now
    given Y such that
A8: x=Y and
A9: Y is being_line and
A10: X '||' Y;
    X // Y by A1,A9,A10,AFF_4:40;
    then Y '||' X by A1,A9,AFF_4:40;
    then
    [Y,X] in {[K,M]: K is being_line & M is being_line & K '||' M} by A1,A9;
    hence x in LDir(X) by A8,EQREL_1:19;
  end;
  hence thesis by A2;
end;

theorem Th10:
  X is being_plane implies for x holds x in PDir(X) iff ex Y st x=
  Y & Y is being_plane & X '||' Y
proof
  assume
A1: X is being_plane;
  let x;
A2: now
    assume x in PDir(X);
    then [x,X] in PlanesParallelity(AS) by EQREL_1:19;
    then consider K,M such that
A3: [x,X]=[K,M] and
A4: K is being_plane and
A5: M is being_plane and
A6: K '||' M;
    take Y=K;
    X=M by A3,XTUPLE_0:1;
    hence x=Y & Y is being_plane & X '||' Y by A3,A4,A5,A6,AFF_4:58,XTUPLE_0:1;
  end;
  now
    given Y such that
A7: x=Y and
A8: Y is being_plane and
A9: X '||' Y;
    Y '||' X by A1,A8,A9,AFF_4:58;
    then [Y,X] in { [K,M]: K is being_plane & M is being_plane & K '||' M} by
A1,A8;
    hence x in PDir(X) by A7,EQREL_1:19;
  end;
  hence thesis by A2;
end;

theorem Th11:
  X is being_line & Y is being_line implies (LDir(X)=LDir(Y) iff X // Y)
proof
  assume that
A1: X is being_line and
A2: Y is being_line;
A3: LDir(Y)= Class(LinesParallelity(AS),Y);
A4: Y in AfLines(AS) by A2;
A5: now
    assume LDir(X)=LDir(Y);
    then X in Class(LinesParallelity(AS),Y) by A4,EQREL_1:23;
    then ex Y9 st X=Y9 & Y9 is being_line & Y '||' Y9 by A2,A3,Th9;
    hence X // Y by A2,AFF_4:40;
  end;
A6: LDir(X)=Class(LinesParallelity(AS),X);
A7: X in AfLines(AS) by A1;
  now
    assume X // Y;
    then X '||' Y by A1,A2,AFF_4:40;
    then Y in Class(LinesParallelity(AS),X) by A1,A2,A6,Th9;
    hence LDir(X)=LDir(Y) by A7,EQREL_1:23;
  end;
  hence thesis by A5;
end;

theorem Th12:
  X is being_line & Y is being_line implies (LDir(X)=LDir(Y) iff X '||' Y)
proof
  assume that
A1: X is being_line and
A2: Y is being_line;
  LDir(X)=LDir(Y) iff X // Y by A1,A2,Th11;
  hence thesis by A1,A2,AFF_4:40;
end;

theorem Th13:
  X is being_plane & Y is being_plane implies (PDir(X)=PDir(Y) iff X '||' Y)
proof
  assume that
A1: X is being_plane and
A2: Y is being_plane;
A3: PDir(Y)= Class(PlanesParallelity(AS),Y);
A4: Y in AfPlanes(AS) by A2;
A5: now
    assume PDir(X)=PDir(Y);
    then X in Class(PlanesParallelity(AS),Y) by A4,EQREL_1:23;
    then ex Y9 st X=Y9 & Y9 is being_plane & Y '||' Y9 by A2,A3,Th10;
    hence X '||' Y by A2,AFF_4:58;
  end;
A6: PDir(X)=Class(PlanesParallelity(AS),X);
A7: X in AfPlanes(AS) by A1;
  now
    assume X '||' Y;
    then Y in Class(PlanesParallelity(AS),X) by A1,A2,A6,Th10;
    hence PDir(X)=PDir(Y) by A7,EQREL_1:23;
  end;
  hence thesis by A5;
end;

definition
  let AS;
  func Dir_of_Lines(AS) -> non empty set equals
  Class LinesParallelity(AS);
  coherence
  proof
    consider a,b being Element of AS such that
A1: a<>b by DIRAF:40;
    set A=Line(a,b);
    A is being_line by A1,AFF_1:def 3;
    then A in AfLines(AS);
    then (Class(LinesParallelity(AS),A)) in Class LinesParallelity(AS) by
EQREL_1:def 3;
    hence thesis;
  end;
end;

definition
  let AS;
  func Dir_of_Planes(AS) -> non empty set equals
  Class PlanesParallelity(AS);
  coherence
  proof
    set a = the Element of AS;
    consider A such that
    a in A and
    a in A and
    a in A and
A1: A is being_plane by AFF_4:37;
    A in AfPlanes(AS) by A1;
    then (Class(PlanesParallelity(AS),A)) in Class PlanesParallelity(AS) by
EQREL_1:def 3;
    hence thesis;
  end;
end;

theorem Th14:
  for x holds x in Dir_of_Lines(AS) iff ex X st x=LDir(X) & X is being_line
proof
  let x;
A1: now
    assume
A2: x in Dir_of_Lines(AS);
    then reconsider x99=x as Subset of AfLines(AS);
    consider x9 being object such that
A3: x9 in AfLines(AS) and
A4: x99=Class(LinesParallelity(AS),x9) by A2,EQREL_1:def 3;
    consider X such that
A5: x9=X and
A6: X is being_line by A3;
    take X;
    thus x=LDir(X) by A4,A5;
    thus X is being_line by A6;
  end;
  now
    given X such that
A7: x=LDir(X) and
A8: X is being_line;
    X in AfLines(AS) by A8;
    hence x in Dir_of_Lines(AS) by A7,EQREL_1:def 3;
  end;
  hence thesis by A1;
end;

theorem Th15:
  for x holds x in Dir_of_Planes(AS) iff ex X st x=PDir(X) & X is being_plane
proof
  let x;
A1: now
    assume
A2: x in Dir_of_Planes(AS);
    then reconsider x99= x as Subset of AfPlanes(AS);
    consider x9 being object such that
A3: x9 in AfPlanes(AS) and
A4: x99=Class(PlanesParallelity(AS),x9) by A2,EQREL_1:def 3;
    consider X such that
A5: x9=X and
A6: X is being_plane by A3;
    take X;
    thus x=PDir(X) by A4,A5;
    thus X is being_plane by A6;
  end;
  now
    given X such that
A7: x=PDir(X) and
A8: X is being_plane;
    X in AfPlanes(AS) by A8;
    hence x in Dir_of_Planes(AS) by A7,EQREL_1:def 3;
  end;
  hence thesis by A1;
end;

:: The point is to guarantee that the classes of new objects consist of really
:: new objects. Clearly the set of directions of lines and the set of affine
:: points do not intersect. However we cannot claim, in general, that the set
:: of affine lines and the set of directions of planes do not intersect; this
:: is evidently true only in the case of affine planes. Therefore we have to
:: modify (slightly) a construction of the set of lines of the projective
:: closure of affine space, when compared with (naive) intuitions.

theorem Th16:
  the carrier of AS misses Dir_of_Lines(AS)
proof
  assume not thesis;
  then consider x being object such that
A1: x in (the carrier of AS) and
A2: x in Dir_of_Lines(AS) by XBOOLE_0:3;
  reconsider a=x as Element of AS by A1;
  consider X such that
A3: x=LDir(X) and
A4: X is being_line by A2,Th14;
  set A=a*X;
A5: A is being_line by A4,AFF_4:27;
  X // A by A4,AFF_4:def 3;
  then X '||' A by A4,A5,AFF_4:40;
  then A in a by A3,A4,A5,Th9;
  hence contradiction by A4,AFF_4:def 3;
end;

theorem
  AS is AffinPlane implies AfLines(AS) misses Dir_of_Planes(AS)
proof
  the carrier of AS c= the carrier of AS;
  then reconsider X9=the carrier of AS as Subset of AS;
  assume AS is AffinPlane;
  then
A1: X9 is being_plane by Th1;
  assume not thesis;
  then consider x being object such that
A2: x in AfLines(AS) and
A3: x in Dir_of_Planes(AS) by XBOOLE_0:3;
  consider Y such that
A4: x=Y and
A5: Y is being_line by A2;
  consider X such that
A6: x=PDir(X) and
A7: X is being_plane by A3,Th15;
  consider a,b being Element of AS such that
A8: a in Y and
  b in Y and
  a<>b by A5,AFF_1:19;
  consider Y9 such that
A9: a = Y9 and
A10: Y9 is being_plane and
  X '||' Y9 by A6,A7,A4,A8,Th10;
A11: not Y9 in Y9;
  Y9 = X9 by A1,A10,AFF_4:33;
  hence contradiction by A9,A11;
end;

theorem Th18:
  for x holds (x in [:AfLines(AS),{1}:] iff ex X st x=[X,1] & X is being_line)
proof
  let x;
A1: now
    assume x in [:AfLines(AS),{1}:];
    then consider x1,x2 being object such that
A2: x1 in AfLines(AS) and
A3: x2 in {1} and
A4: x=[x1,x2] by ZFMISC_1:def 2;
    consider X such that
A5: x1=X and
A6: X is being_line by A2;
    take X;
    thus x=[X,1] by A3,A4,A5,TARSKI:def 1;
    thus X is being_line by A6;
  end;
  now
    given X such that
A7: x=[X,1] and
A8: X is being_line;
    X in AfLines(AS ) by A8;
    hence x in [:AfLines(AS),{1}:] by A7,ZFMISC_1:106;
  end;
  hence thesis by A1;
end;

theorem Th19:
  for x holds (x in [:Dir_of_Planes(AS),{2}:] iff ex X st x=[PDir(
  X),2] & X is being_plane)
proof
  let x;
A1: now
    assume x in [:Dir_of_Planes(AS),{2}:];
    then consider x1,x2 being object such that
A2: x1 in Dir_of_Planes(AS) and
A3: x2 in {2} and
A4: x=[x1,x2] by ZFMISC_1:def 2;
    consider X such that
A5: x1=PDir(X) and
A6: X is being_plane by A2,Th15;
    take X;
    thus x=[PDir(X),2] by A3,A4,A5,TARSKI:def 1;
    thus X is being_plane by A6;
  end;
   (ex X st x=[PDir(X),2] & X is being_plane)
      implies x in [:Dir_of_Planes(AS),{2}:] by Th15,ZFMISC_1:106;
  hence thesis by A1;
end;

definition
  let AS;
  func ProjectivePoints(AS) -> non empty set equals
  (the carrier of AS) \/
  Dir_of_Lines(AS);
  correctness;
end;

definition
  let AS;
  func ProjectiveLines(AS) -> non empty set equals
  [:AfLines(AS),{1}:] \/ [:
  Dir_of_Planes(AS),{2}:];
  coherence;
end;

definition
  let AS;
  func Proj_Inc(AS) -> Relation of ProjectivePoints(AS),ProjectiveLines(AS)
  means
  :Def11:
  for x,y being object
    holds [x,y] in it iff (ex K st K is being_line & y=[K,1]
  & (x in the carrier of AS & x in K or x = LDir(K))) or ex K,X st K is
  being_line & X is being_plane & x=LDir(K) & y=[PDir(X),2] & K '||' X;
  existence
  proof
    defpred P[object,object] means
    ((ex K st K is being_line & $2=[K,1] & ($1 in the
carrier of AS & $1 in K or $1 = LDir(K))) or (ex K,X st K is being_line & X is
    being_plane & $1=LDir(K) & $2=[PDir(X),2] & K '||' X));
    set VV1 = ProjectivePoints(AS), VV2 = ProjectiveLines(AS);
    consider P being Relation of VV1,VV2 such that
A1: for x,y being object holds [x,y] in P iff x in VV1 & y in VV2 & P[x,y]
     from RELSET_1:sch 1;
    take P;
    let x,y be object;
    thus [x,y] in P implies (ex K st K is being_line & y=[K,1] & (x in the
    carrier of AS & x in K or x = LDir(K))) or ex K,X st K is being_line & X is
    being_plane & x=LDir(K) & y=[PDir(X),2] & K '||' X by A1;
    assume
A2: (ex K st K is being_line & y=[K,1] & (x in the carrier of AS & x
in K or x = LDir(K))) or ex K,X st K is being_line & X is being_plane & x=LDir(
    K) & y=[PDir(X),2] & K '||' X;
    x in VV1 & y in VV2
    proof
A3:   now
        given K such that
A4:     K is being_line and
A5:     y=[K,1] and
A6:     x in the carrier of AS & x in K or x = LDir(K);
A7:     now
          assume x=LDir(K);
          then x in Dir_of_Lines(AS) by A4,Th14;
          hence x in VV1 by XBOOLE_0:def 3;
        end;
        y in [:AfLines(AS),{1}:] by A4,A5,Th18;
        hence thesis by A6,A7,XBOOLE_0:def 3;
      end;
      now
        given K,X such that
A8:     K is being_line and
A9:     X is being_plane and
A10:    x=LDir(K) and
A11:    y=[PDir(X),2] and
        K '||' X;
        x in Dir_of_Lines(AS) by A8,A10,Th14;
        hence x in VV1 by XBOOLE_0:def 3;
        y in [:Dir_of_Planes(AS),{2}:] by A9,A11,Th19;
        hence y in VV2 by XBOOLE_0:def 3;
      end;
      hence thesis by A2,A3;
    end;
    hence thesis by A1,A2;
  end;
  uniqueness
  proof
    let P,Q be Relation of ProjectivePoints(AS),ProjectiveLines(AS) such that
A12: for x,y being object holds
     [x,y] in P iff (ex K st K is being_line & y=[K,1] & (x in the
    carrier of AS & x in K or x = LDir(K))) or ex K,X st K is being_line & X is
    being_plane & x=LDir(K) & y=[PDir(X),2] & K '||' X and
A13: for x,y being object holds
     [x,y] in Q iff (ex K st K is being_line & y=[K,1] & (x in the
    carrier of AS & x in K or x = LDir(K))) or ex K,X st K is being_line & X is
    being_plane & x=LDir(K) & y=[PDir(X),2] & K '||' X;
    for x,y being object holds [x,y] in P iff [x,y] in Q
    proof
      let x,y be object;
      [x,y] in P iff (ex K st K is being_line & y=[K,1] & (x in the
carrier of AS & x in K or x = LDir(K))) or ex K,X st K is being_line & X is
      being_plane & x=LDir(K) & y=[PDir(X),2] & K '||' X by A12;
      hence thesis by A13;
    end;
    hence thesis by RELAT_1:def 2;
  end;
end;

definition
  let AS;
  func Inc_of_Dir(AS) -> Relation of Dir_of_Lines(AS),Dir_of_Planes(AS) means
  :Def12:
  for x,y being object
    holds [x,y] in it iff ex A,X st x=LDir(A) & y=PDir(X) & A is
  being_line & X is being_plane & A '||' X;
  existence
  proof
    defpred P[object,object] means ex A,X st $1=LDir(A) & $2=PDir(X) & A is
    being_line & X is being_plane & A '||' X;
    set VV1 = Dir_of_Lines(AS), VV2 = Dir_of_Planes(AS);
    consider P being Relation of VV1,VV2 such that
A1: for x,y being object holds [x,y] in P iff x in VV1 & y in VV2 & P[x,y]
from RELSET_1:sch 1;
    take P;
    let x,y be object;
    thus [x,y] in P implies ex A,X st x=LDir(A) & y=PDir(X) & A is being_line
    & X is being_plane & A '||' X by A1;
    assume
A2: ex A,X st x=LDir(A) & y=PDir(X) & A is being_line & X is
    being_plane & A '||' X;
    then
A3: y in VV2 by Th15;
    x in VV1 by A2,Th14;
    hence thesis by A1,A2,A3;
  end;
  uniqueness
  proof
    let P,Q be Relation of Dir_of_Lines(AS),Dir_of_Planes(AS) such that
A4: for x,y being object holds
     [x,y] in P iff ex A,X st x=LDir(A) & y=PDir(X) & A is being_line &
    X is being_plane & A '||' X and
A5: for x,y being object holds
     [x,y] in Q iff ex A,X st x=LDir(A) & y=PDir(X) & A is being_line &
    X is being_plane & A '||' X;
    for x,y being object holds [x,y] in P iff [x,y] in Q
    proof
      let x,y be object;
      [x,y] in P iff ex A,X st x=LDir(A) & y=PDir(X) & A is being_line &
      X is being_plane & A '||' X by A4;
      hence thesis by A5;
    end;
    hence thesis by RELAT_1:def 2;
  end;
end;

definition
  let AS;
  func IncProjSp_of(AS) -> strict IncProjStr equals
  IncProjStr (#
    ProjectivePoints(AS), ProjectiveLines(AS), Proj_Inc(AS) #);
  correctness;
end;

definition
  let AS;
  func ProjHorizon(AS) -> strict IncProjStr equals
  IncProjStr (#Dir_of_Lines(
    AS), Dir_of_Planes(AS), Inc_of_Dir(AS) #);
  correctness;
end;

theorem Th20:
  for x holds (x is POINT of IncProjSp_of(AS) iff (x is Element of
  AS or ex X st x=LDir(X) & X is being_line))
proof
  let x;
A1: now
A2: now
      given X such that
A3:   x=LDir(X) and
A4:   X is being_line;
      x in Dir_of_Lines( AS ) by A3,A4,Th14;
      hence x is POINT of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    assume x is Element of AS or ex X st x=LDir(X) & X is being_line;
    hence x is POINT of IncProjSp_of(AS) by A2,XBOOLE_0:def 3;
  end;
  now
    assume
A5: x is POINT of IncProjSp_of(AS);
    x in Dir_of_Lines(AS) implies ex X st x=LDir(X) & X is being_line by Th14;
    hence x is Element of AS or ex X st x=LDir(X) & X is being_line by A5,
XBOOLE_0:def 3;
  end;
  hence thesis by A1;
end;

theorem
  x is Element of the Points of ProjHorizon(AS) iff ex X st x=LDir(X) &
  X is being_line by Th14;

theorem Th22:
  x is Element of the Points of ProjHorizon(AS) implies x is POINT
  of IncProjSp_of(AS)
proof
  assume x is Element of the Points of ProjHorizon(AS);
  then ex X st x=LDir(X) & X is being_line by Th14;
  hence thesis by Th20;
end;

theorem Th23:
  for x holds (x is LINE of IncProjSp_of(AS) iff ex X st (x=[X,1]
  & X is being_line or x=[PDir(X),2] & X is being_plane))
proof
  let x;
A1: now
    given X such that
A2: x=[X,1] & X is being_line or x=[PDir(X),2] & X is being_plane;
A3: now
      assume that
A4:   x=[PDir(X),2] and
A5:   X is being_plane;
      x in [:Dir_of_Planes(AS),{2}:] by A4,A5,Th19;
      hence x is LINE of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    now
      assume that
A6:   x=[X,1] and
A7:   X is being_line;
      x in [:AfLines(AS),{1}:] by A6,A7,Th18;
      hence x is LINE of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    hence x is LINE of IncProjSp_of(AS) by A2,A3;
  end;
  now
A8: x in [:Dir_of_Planes(AS),{2}:] implies ex X st x=[X,1] & X is
    being_line or x=[PDir(X),2] & X is being_plane by Th19;
    assume
A9: x is LINE of IncProjSp_of(AS);
    x in [:AfLines(AS),{1}:] implies ex X st x=[X,1] & X is being_line or
    x=[PDir(X),2] & X is being_plane by Th18;
    hence
    ex X st x=[X,1] & X is being_line or x=[PDir(X),2] & X is being_plane
    by A9,A8,XBOOLE_0:def 3;
  end;
  hence thesis by A1;
end;

theorem
  x is Element of the Lines of ProjHorizon(AS) iff ex X st x=PDir(X) & X
  is being_plane by Th15;

theorem Th25:
  x is Element of the Lines of ProjHorizon(AS) implies [x,2] is
  LINE of IncProjSp_of(AS)
proof
  assume x is Element of the Lines of ProjHorizon(AS);
  then ex X st x=PDir(X) & X is being_plane by Th15;
  hence thesis by Th23;
end;

reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);

theorem Th26:
  x=a & [X,1]=A implies (a on A iff X is being_line & x in X)
proof
  assume that
A1: x=a and
A2: [X,1]=A;
A3: now
A4: now
      given K such that
A5:   K is being_line and
A6:   [X,1]=[K,1] and
A7:   x in the carrier of AS & x in K or x = LDir(K);
A8:   now
        assume x=LDir(K);
        then x in Dir_of_Lines(AS) by A5,Th14;
        then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by XBOOLE_0:def 4;
        then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
        hence contradiction by Th16;
      end;
      X=[K,1]`1 by A6
        .= K;
      hence X is being_line & x in X by A5,A7,A8;
    end;
    assume a on A;
    then
A9: [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
    not ex K,X9 st K is being_line & X9 is being_plane & x=LDir(K) & [X,1
    ]=[PDir(X9),2] & K '||' X9 by XTUPLE_0:1;
    hence X is being_line & x in X by A1,A2,A9,A4,Def11;
  end;
  now
    assume that
A10: X is being_line and
A11: x in X;
    [x,[X,1]] in Proj_Inc(AS) by A10,A11,Def11;
    hence a on A by A1,A2,INCSP_1:def 1;
  end;
  hence thesis by A3;
end;

theorem Th27:
  x=a & [PDir(X),2]=A implies not a on A
proof
  assume that
A1: x=a and
A2: [PDir(X),2]=A;
A3: now
    given K such that
    K is being_line and
A4: [PDir(X),2]=[K,1] and
    x in the carrier of AS & x in K or x = LDir(K);
    2 = [K,1]`2 by A4
      .= 1;
    hence contradiction;
  end;
A5: now
    given K,X9 such that
A6: K is being_line and
    X9 is being_plane and
A7: x=LDir(K) and
    [PDir(X),2]=[PDir(X9),2] and
    K '||' X9;
    x in Dir_of_Lines(AS) by A6,A7,Th14;
    then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by XBOOLE_0:def 4;
    then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
    hence contradiction by Th16;
  end;
  assume a on A;
  then [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
  hence contradiction by A1,A2,A3,A5,Def11;
end;

theorem Th28:
  a=LDir(Y) & [X,1]=A & Y is being_line & X is being_line implies
  (a on A iff Y '||' X)
proof
  assume that
A1: a=LDir(Y) and
A2: [X,1]=A and
A3: Y is being_line and
A4: X is being_line;
A5: now
A6: now
      given K such that
A7:   K is being_line and
A8:   [X,1]=[K,1] and
A9:   LDir(Y) in the carrier of AS & LDir(Y) in K or LDir(Y) = LDir(K );
A10:  K in AfLines(AS) by A7;
A11:  X=K by A8,XTUPLE_0:1;
A12:  now
        assume LDir(Y)=LDir(K);
        then
A13:    Y in Class(LinesParallelity(AS),K) by A10,EQREL_1:23;
        LDir(K)=Class(LinesParallelity(AS),K);
        then consider K9 being Subset of AS such that
A14:    Y=K9 and
A15:    K9 is being_line and
A16:    K '||' K9 by A7,A13,Th9;
        K // K9 by A7,A15,A16,AFF_4:40;
        hence Y '||' X by A7,A11,A14,A15,AFF_4:40;
      end;
      now
        assume that
A17:    LDir(Y) in the carrier of AS and
        LDir(Y) in K;
        a in Dir_of_Lines(AS) by A1,A3,Th14;
        then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by A1,A17,
XBOOLE_0:def 4;
        then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
        hence contradiction by Th16;
      end;
      hence Y '||' X by A9,A12;
    end;
    assume a on A;
    then
A18: [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
    not ex K,X9 st K is being_line & X9 is being_plane & LDir(Y)=LDir(K)
    & [X,1]=[PDir(X9),2] & K '||' X9 by XTUPLE_0:1;
    hence Y '||' X by A1,A2,A18,A6,Def11;
  end;
  now
    assume Y '||' X;
    then
A19: X in LDir(Y) by A3,A4,Th9;
A20: LDir(X)=Class(LinesParallelity(AS),X);
    Y in AfLines(AS) by A3;
    then Class(LinesParallelity(AS),X)=Class(LinesParallelity(AS),Y) by A19,
EQREL_1:23;
    then [a,A] in Proj_Inc(AS) by A1,A2,A4,A20,Def11;
    hence a on A by INCSP_1:def 1;
  end;
  hence thesis by A5;
end;

theorem Th29:
  a=LDir(Y) & A=[PDir(X),2] & Y is being_line & X is being_plane
  implies (a on A iff Y '||' X)
proof
  assume that
A1: a=LDir(Y) and
A2: A=[PDir(X),2] and
A3: Y is being_line and
A4: X is being_plane;
A5: now
A6: now
      given K,X9 such that
A7:   K is being_line and
A8:   X9 is being_plane and
A9:   LDir(Y)=LDir(K) and
A10:  [PDir(X),2]=[PDir(X9),2] and
A11:  K '||' X9;
A12:  X9 in AfPlanes(AS) by A8;
A13:  Class(PlanesParallelity(AS),X9)= PDir(X9);
      PDir(X)=PDir(X9) by A10,XTUPLE_0:1;
      then X in Class(PlanesParallelity(AS),X9) by A12,EQREL_1:23;
      then
A14:  ex X99 being Subset of AS st X=X99 & X99 is being_plane & X9 '||'
      X99 by A8,A13,Th10;
      K in AfLines(AS) by A7;
      then
A15:  Y in Class(LinesParallelity(AS),K) by A9,EQREL_1:23;
      Class(LinesParallelity(AS),K)= LDir(K);
      then consider K9 being Subset of AS such that
A16:  Y=K9 and
A17:  K9 is being_line and
A18:  K '||' K9 by A7,A15,Th9;
      K // K9 by A7,A17,A18,AFF_4:40;
      then K9 '||' X9 by A11,AFF_4:56;
      hence Y '||' X by A8,A16,A14,AFF_4:59,60;
    end;
    assume a on A;
    then
A19: [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
    (ex K st K is being_line & [PDir(X),2]=[K,1] & (LDir(Y) in the carrier
of AS & LDir(Y) in K or LDir(Y) = LDir(K))) implies contradiction by XTUPLE_0:1
;
    hence Y '||' X by A1,A2,A19,A6,Def11;
  end;
  now
    assume Y '||' X;
    then [LDir(Y),[PDir(X),2]] in Proj_Inc(AS) by A3,A4,Def11;
    hence a on A by A1,A2,INCSP_1:def 1;
  end;
  hence thesis by A5;
end;

theorem Th30:
  X is being_line & a=LDir(X) & A=[X,1] implies a on A
proof
  assume that
A1: X is being_line and
A2: a=LDir(X) and
A3: A=[X,1];
  X // X by A1,AFF_1:41;
  then X '||' X by A1,AFF_4:40;
  hence thesis by A1,A2,A3,Th28;
end;

theorem Th31:
  X is being_line & Y is being_plane & X c= Y & a=LDir(X) & A=[
  PDir(Y),2] implies a on A
proof
  assume that
A1: X is being_line and
A2: Y is being_plane and
A3: X c= Y and
A4: a=LDir(X) and
A5: A=[PDir(Y),2];
  X '||' Y by A1,A2,A3,AFF_4:42;
  hence thesis by A1,A2,A4,A5,Th29;
end;

theorem Th32:
  Y is being_plane & X c= Y & X9 // X & a=LDir(X9) & A=[PDir(Y),2]
  implies a on A
proof
  assume that
A1: Y is being_plane and
A2: X c= Y and
A3: X9 // X and
A4: a=LDir(X9) and
A5: A=[PDir(Y),2];
  X is being_line by A3,AFF_1:36;
  then
A6: X9 '||' Y by A1,A2,A3,AFF_4:42,56;
  X9 is being_line by A3,AFF_1:36;
  hence thesis by A1,A4,A5,A6,Th29;
end;

theorem
  A=[PDir(X),2] & X is being_plane & a on A implies a is not Element of
  AS by Th27;

theorem Th34:
  A=[X,1] & X is being_line & p on A & p is not Element of AS implies p=LDir(X)
proof
  assume that
A1: A=[X,1] and
A2: X is being_line and
A3: p on A and
A4: not p is Element of AS;
  consider Xp being Subset of AS such that
A5: p=LDir(Xp) and
A6: Xp is being_line by A4,Th20;
  Xp '||' X by A1,A2,A3,A5,A6,Th28;
  hence thesis by A2,A5,A6,Th12;
end;

theorem Th35:
  A=[X,1] & X is being_line & p on A & a on A & a<>p & not p is
  Element of AS implies a is Element of AS
proof
  assume that
A1: A=[X,1] and
A2: X is being_line and
A3: p on A and
A4: a on A and
A5: a<>p and
A6: not p is Element of AS;
  assume not thesis;
  then a=LDir(X) by A1,A2,A4,Th34;
  hence contradiction by A1,A2,A3,A5,A6,Th34;
end;

theorem Th36:
  for a being Element of the Points of ProjHorizon(AS),A being
  Element of the Lines of ProjHorizon(AS) st a=LDir(X) & A=PDir(Y) & X is
  being_line & Y is being_plane holds (a on A iff X '||' Y)
proof
  let a be Element of the Points of ProjHorizon(AS),A be Element of the Lines
  of ProjHorizon(AS) such that
A1: a=LDir(X) and
A2: A=PDir(Y) and
A3: X is being_line and
A4: Y is being_plane;
A5: now
    assume a on A;
    then [a,A] in the Inc of ProjHorizon(AS) by INCSP_1:def 1;
    then consider X9,Y9 such that
A6: a=LDir(X9) and
A7: A=PDir(Y9) and
A8: X9 is being_line and
A9: Y9 is being_plane and
A10: X9 '||' Y9 by Def12;
    X // X9 by A1,A3,A6,A8,Th11;
    then
A11: X '||' Y9 by A10,AFF_4:56;
    Y9 '||' Y by A2,A4,A7,A9,Th13;
    hence X '||' Y by A9,A11,AFF_4:59,60;
  end;
  now
    assume X '||' Y;
    then [a,A] in Inc_of_Dir(AS) by A1,A2,A3,A4,Def12;
    hence a on A by INCSP_1:def 1;
  end;
  hence thesis by A5;
end;

theorem Th37:
  for a being Element of the Points of ProjHorizon(AS),a9 being
  Element of the Points of IncProjSp_of(AS),A being Element of the Lines of
  ProjHorizon(AS),A9 being LINE of IncProjSp_of(AS) st a9=a & A9=[A,2] holds (a
  on A iff a9 on A9)
proof
  let a be Element of the Points of ProjHorizon(AS),a9 be Element of the
  Points of IncProjSp_of(AS),A be Element of the Lines of ProjHorizon(AS),A9 be
  LINE of IncProjSp_of(AS) such that
A1: a9=a and
A2: A9=[A,2];
  consider X such that
A3: a=LDir(X) and
A4: X is being_line by Th14;
  consider Y such that
A5: A=PDir(Y) and
A6: Y is being_plane by Th15;
A7: now
    assume a9 on A9;
    then X '||' Y by A1,A2,A3,A4,A5,A6,Th29;
    hence a on A by A3,A4,A5,A6,Th36;
  end;
  now
    assume a on A;
    then X '||' Y by A3,A4,A5,A6,Th36;
    hence a9 on A9 by A1,A2,A3,A4,A5,A6,Th29;
  end;
  hence thesis by A7;
end;

reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem Th38:
  for a,b being Element of the Points of ProjHorizon(AS), A,K
  being Element of the Lines of ProjHorizon(AS) st a on A & a on K & b on A & b
  on K holds a=b or A=K
proof
  let a,b be Element of the Points of ProjHorizon(AS), A,K be Element of the
  Lines of ProjHorizon(AS) such that
A1: a on A and
A2: a on K and
A3: b on A and
A4: b on K;
  consider Y9 such that
A5: b=LDir(Y9) and
A6: Y9 is being_line by Th14;
  consider X9 such that
A7: K=PDir(X9) and
A8: X9 is being_plane by Th15;
A9: Y9 '||' X9 by A4,A5,A6,A7,A8,Th36;
  consider Y such that
A10: a=LDir(Y) and
A11: Y is being_line by Th14;
  assume a<>b;
  then
A12: not Y // Y9 by A10,A11,A5,A6,Th11;
  consider X such that
A13: A=PDir(X) and
A14: X is being_plane by Th15;
A15: Y9 '||' X by A3,A5,A6,A13,A14,Th36;
A16: Y '||' X9 by A2,A10,A11,A7,A8,Th36;
  Y '||' X by A1,A10,A11,A13,A14,Th36;
  then X '||' X9 by A11,A6,A14,A8,A12,A16,A15,A9,Th5;
  hence thesis by A13,A14,A7,A8,Th13;
end;

theorem Th39:
  for A being Element of the Lines of ProjHorizon(AS) ex a,b,c
being Element of the Points of ProjHorizon(AS) st a on A & b on A & c on A & a
  <>b & b<>c & c <>a
proof
  let A be Element of the Lines of ProjHorizon(AS);
  consider X such that
A1: A=PDir(X) and
A2: X is being_plane by Th15;
  consider x,y,z such that
A3: x in X and
A4: y in X and
A5: z in X and
A6: not LIN x,y,z by A2,AFF_4:34;
A7: y<>z by A6,AFF_1:7;
  then
A8: Line(y,z) is being_line by AFF_1:def 3;
  then
A9: Line(y,z) '||' X by A2,A4,A5,A7,AFF_4:19,42;
A10: z<>x by A6,AFF_1:7;
  then
A11: Line(x,z) is being_line by AFF_1:def 3;
  then
A12: Line(x,z) '||' X by A2,A3,A5,A10,AFF_4:19,42;
A13: x<>y by A6,AFF_1:7;
  then
A14: Line(x,y) is being_line by AFF_1:def 3;
  then reconsider
  a=LDir(Line(x,y)),b=LDir(Line(y,z)),c =LDir(Line(x,z)) as Element
  of the Points of ProjHorizon(AS) by A8,A11,Th14;
  take a,b,c;
  Line(x,y) '||' X by A2,A3,A4,A13,A14,AFF_4:19,42;
  hence a on A & b on A & c on A by A1,A2,A14,A8,A11,A9,A12,Th36;
A15: x in Line(x,y) by AFF_1:15;
A16: z in Line(y,z) by AFF_1:15;
A17: y in Line(x,y) by AFF_1:15;
A18: y in Line(y,z) by AFF_1:15;
  thus a<>b
  proof
    assume not thesis;
    then Line(x,y) // Line(y,z) by A14,A8,Th11;
    then z in Line(x,y) by A17,A18,A16,AFF_1:45;
    hence contradiction by A6,A14,A15,A17,AFF_1:21;
  end;
A19: z in Line(x,z) by AFF_1:15;
A20: x in Line(x,z) by AFF_1:15;
  thus b<>c
  proof
    assume not thesis;
    then Line(y,z) // Line(x,z) by A8,A11,Th11;
    then x in Line(y,z) by A16,A20,A19,AFF_1:45;
    hence contradiction by A6,A8,A18,A16,AFF_1:21;
  end;
  thus c <>a
  proof
    assume not thesis;
    then Line(x,y) // Line(x,z) by A14,A11,Th11;
    then z in Line(x,y) by A15,A20,A19,AFF_1:45;
    hence contradiction by A6,A14,A15,A17,AFF_1:21;
  end;
end;

theorem Th40:
  for a,b being Element of the Points of ProjHorizon(AS) ex A
  being Element of the Lines of ProjHorizon(AS) st a on A & b on A
proof
  let a,b be Element of the Points of ProjHorizon(AS);
  consider X such that
A1: a=LDir(X) and
A2: X is being_line by Th14;
  consider X9 such that
A3: b=LDir(X9) and
A4: X9 is being_line by Th14;
  consider x,y being Element of AS such that
A5: x in X9 and
  y in X9 and
  x<>y by A4,AFF_1:19;
A6: x in x*X by A2,AFF_4:def 3;
  x*X is being_line by A2,AFF_4:27;
  then consider Z such that
A7: X9 c= Z and
A8: x*X c= Z and
A9: Z is being_plane by A4,A5,A6,AFF_4:38;
A10: X9 '||' Z by A4,A7,A9,AFF_4:42;
  reconsider A=PDir(Z) as Element of the Lines of ProjHorizon(AS) by A9,Th15;
  take A;
  X // x*X by A2,AFF_4:def 3;
  then X '||' Z by A2,A8,A9,AFF_4:41;
  hence thesis by A1,A2,A3,A4,A9,A10,Th36;
end;

Lm1: AS is not AffinPlane implies ex a being Element of the Points of
ProjHorizon(AS),A being Element of the Lines of ProjHorizon(AS) st not a on A
proof
  set x = the Element of AS;
  consider X such that
A1: x in X and
  x in X and
  x in X and
A2: X is being_plane by AFF_4:37;
  reconsider A=PDir(X) as Element of the Lines of ProjHorizon(AS) by A2,Th15;
  assume AS is not AffinPlane;
  then consider t such that
A3: not t in X by A2,AFF_4:48;
  set Y=Line(x,t);
A4: Y is being_line by A1,A3,AFF_1:def 3;
  then reconsider a=LDir(Y) as Element of the Points of ProjHorizon(AS) by Th14
;
  take a,A;
A5: t in Y by AFF_1:15;
A6: x in Y by AFF_1:15;
  thus not a on A
  proof
    assume not thesis;
    then Y '||' X by A2,A4,Th36;
    then Y c= X by A1,A2,A4,A6,AFF_4:43;
    hence contradiction by A3,A5;
  end;
end;

Lm2: a on A & a on K & b on A & b on K implies a=b or A=K
proof
  assume that
A1: a on A and
A2: a on K and
A3: b on A and
A4: b on K;
  consider X such that
A5: A=[X,1] & X is being_line or A=[PDir(X),2] & X is being_plane by Th23;
  consider X9 such that
A6: K=[X9,1] & X9 is being_line or K=[PDir(X9),2] & X9 is being_plane by Th23;
  assume
A7: a<>b;
A8: now
    given Y such that
A9: a=LDir(Y) and
A10: Y is being_line;
A11: now
      given Y9 such that
A12:  b=LDir(Y9) and
A13:  Y9 is being_line;
A14:  not Y // Y9 by A7,A9,A10,A12,A13,Th11;
A15:  M=[Z,1] & Z is being_line implies not (a on M & b on M)
      proof
        assume that
A16:    M=[Z,1] and
A17:    Z is being_line;
        assume
A18:    not thesis;
        then Y9 '||' Z by A12,A13,A16,A17,Th28;
        then
A19:    Y9 // Z by A13,A17,AFF_4:40;
        Y '||' Z by A9,A10,A16,A17,A18,Th28;
        then Y // Z by A10,A17,AFF_4:40;
        then Y // Y9 by A19,AFF_1:44;
        hence contradiction by A7,A9,A10,A12,A13,Th11;
      end;
      then
A20:  Y9 '||' X by A1,A3,A5,A12,A13,Th29;
A21:  Y9 '||' X9 by A2,A4,A6,A12,A13,A15,Th29;
A22:  Y '||' X9 by A2,A4,A6,A9,A10,A15,Th29;
      Y '||' X by A1,A3,A5,A9,A10,A15,Th29;
      then X '||' X9 by A1,A2,A3,A4,A5,A6,A10,A13,A15,A14,A22,A20,A21,Th5;
      hence thesis by A1,A2,A3,A4,A5,A6,A15,Th13;
    end;
    now
      assume b is Element of AS;
      then reconsider y=b as Element of AS;
A23:  y in X9 by A4,A6,Th26,Th27;
A24:  y=b;
      then Y '||' X9 by A2,A4,A6,A9,A10,Th27,Th28;
      then
A25:  Y // X9 by A4,A6,A10,A24,Th27,AFF_4:40;
      Y '||' X by A1,A3,A5,A9,A10,A24,Th27,Th28;
      then Y // X by A3,A5,A10,A24,Th27,AFF_4:40;
      then
A26:  X // X9 by A25,AFF_1:44;
      y in X by A3,A5,Th26,Th27;
      hence thesis by A3,A4,A5,A6,A23,A26,Th27,AFF_1:45;
    end;
    hence thesis by A11,Th20;
  end;
  now
    assume a is Element of AS;
    then reconsider x=a as Element of AS;
A27: x=a;
A28: x in X9 by A2,A6,Th26,Th27;
A29: x in X by A1,A5,Th26,Th27;
A30: now
      given Y such that
A31:  b=LDir(Y) and
A32:  Y is being_line;
      Y '||' X9 by A2,A4,A6,A27,A31,A32,Th27,Th28;
      then
A33:  Y // X9 by A2,A6,A27,A32,Th27,AFF_4:40;
      Y '||' X by A1,A3,A5,A27,A31,A32,Th27,Th28;
      then Y // X by A1,A5,A27,A32,Th27,AFF_4:40;
      then X // X9 by A33,AFF_1:44;
      hence thesis by A1,A2,A5,A6,A29,A28,Th27,AFF_1:45;
    end;
    now
      assume b is Element of AS;
      then reconsider y=b as Element of AS;
A34:  y in X9 by A4,A6,Th26,Th27;
      y in X by A3,A5,Th26,Th27;
      hence thesis by A1,A2,A7,A5,A6,A29,A28,A34,Th27,AFF_1:18;
    end;
    hence thesis by A30,Th20;
  end;
  hence thesis by A8,Th20;
end;

Lm3: ex a,b,c st a on A & b on A & c on A & a<>b & b<>c & c <>a
proof
  consider X such that
A1: A=[X,1] & X is being_line or A=[PDir(X),2] & X is being_plane by Th23;
A2: now
    assume that
A3: A=[PDir(X),2] and
A4: X is being_plane;
    consider x,y,z such that
A5: x in X and
A6: y in X and
A7: z in X and
A8: not LIN x,y,z by A4,AFF_4:34;
A9: y<>z by A8,AFF_1:7;
    then
A10: Line(y,z) is being_line by AFF_1:def 3;
    then
A11: Line(y,z) '||' X by A4,A6,A7,A9,AFF_4:19,42;
A12: z<>x by A8,AFF_1:7;
    then
A13: Line(x,z) is being_line by AFF_1:def 3;
    then
A14: Line(x,z) '||' X by A4,A5,A7,A12,AFF_4:19,42;
A15: x<>y by A8,AFF_1:7;
    then
A16: Line(x,y) is being_line by AFF_1:def 3;
    then reconsider
    a=LDir(Line(x,y)),b=LDir(Line(y,z)),c =LDir(Line(x,z)) as POINT
    of IncProjSp_of(AS) by A10,A13,Th20;
    take a,b,c;
    Line(x,y) '||' X by A4,A5,A6,A15,A16,AFF_4:19,42;
    hence a on A & b on A & c on A by A3,A4,A16,A10,A13,A11,A14,Th29;
A17: x in Line(x,y) by AFF_1:15;
A18: z in Line(y,z) by AFF_1:15;
A19: y in Line(x,y) by AFF_1:15;
A20: y in Line(y,z) by AFF_1:15;
    thus a<>b
    proof
      assume not thesis;
      then Line(x,y) // Line(y,z) by A16,A10,Th11;
      then z in Line(x,y) by A19,A20,A18,AFF_1:45;
      hence contradiction by A8,A16,A17,A19,AFF_1:21;
    end;
A21: z in Line(x,z) by AFF_1:15;
A22: x in Line(x,z) by AFF_1:15;
    thus b<>c
    proof
      assume not thesis;
      then Line(y,z) // Line(x,z) by A10,A13,Th11;
      then x in Line(y,z) by A18,A22,A21,AFF_1:45;
      hence contradiction by A8,A10,A20,A18,AFF_1:21;
    end;
    thus c <>a
    proof
      assume not thesis;
      then Line(x,y) // Line(x,z) by A16,A13,Th11;
      then z in Line(x,y) by A17,A22,A21,AFF_1:45;
      hence contradiction by A8,A16,A17,A19,AFF_1:21;
    end;
  end;
  now
    assume that
A23: A=[X,1] and
A24: X is being_line;
    reconsider c =LDir(X) as POINT of IncProjSp_of(AS) by A24,Th20;
    consider x,y such that
A25: x in X and
A26: y in X and
A27: x<>y by A24,AFF_1:19;
    reconsider a=x,b=y as Element of the Points of IncProjSp_of(AS) by Th20;
    take a,b,c;
    X // X by A24,AFF_1:41;
    then X '||' X by A24,AFF_4:40;
    hence a on A & b on A & c on A by A23,A24,A25,A26,Th26,Th28;
    thus a<>b by A27;
    thus b<>c & c <>a
    proof
      assume
A28:  not thesis;
      c in Dir_of_Lines(AS) by A24,Th14;
      then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by A28,XBOOLE_0:def 4;
      then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
      hence contradiction by Th16;
    end;
  end;
  hence thesis by A1,A2;
end;

Lm4: ex A st a on A & b on A
proof
A1: now
    given X such that
A2: a=LDir(X) and
A3: X is being_line;
A4: now
      given X9 such that
A5:   b=LDir(X9) and
A6:   X9 is being_line;
      consider x,y being Element of AS such that
A7:   x in X9 and
      y in X9 and
      x<>y by A6,AFF_1:19;
A8:   x in x*X by A3,AFF_4:def 3;
      x*X is being_line by A3,AFF_4:27;
      then consider Z such that
A9:   X9 c= Z and
A10:  x*X c= Z and
A11:  Z is being_plane by A6,A7,A8,AFF_4:38;
A12:  X9 '||' Z by A6,A9,A11,AFF_4:42;
      reconsider A=[PDir(Z),2] as LINE of IncProjSp_of(AS) by A11,Th23;
      take A;
      X // x*X by A3,AFF_4:def 3;
      then X '||' Z by A3,A10,A11,AFF_4:41;
      hence a on A & b on A by A2,A3,A5,A6,A11,A12,Th29;
    end;
    now
      assume b is Element of AS;
      then reconsider y=b as Element of AS;
A13:  y*X is being_line by A3,AFF_4:27;
      then reconsider A=[y*X,1] as LINE of IncProjSp_of(AS) by Th23;
      take A;
      X // y*X by A3,AFF_4:def 3;
      then X '||' y*X by A3,A13,AFF_4:40;
      hence a on A by A2,A3,A13,Th28;
      y in y*X by A3,AFF_4:def 3;
      hence b on A by A13,Th26;
    end;
    hence thesis by A4,Th20;
  end;
  now
    assume a is Element of AS;
    then reconsider x=a as Element of AS;
A14: now
      given X9 such that
A15:  b=LDir(X9) and
A16:  X9 is being_line;
A17:  x*X9 is being_line by A16,AFF_4:27;
      then reconsider A=[x*X9,1] as LINE of IncProjSp_of(AS) by Th23;
      take A;
      x in x*X9 by A16,AFF_4:def 3;
      hence a on A by A17,Th26;
      X9 // x*X9 by A16,AFF_4:def 3;
      then X9 '||' x*X9 by A16,A17,AFF_4:40;
      hence b on A by A15,A16,A17,Th28;
    end;
    now
      assume b is Element of AS;
      then reconsider y=b as Element of AS;
      consider Y such that
A18:  x in Y and
A19:  y in Y and
A20:  Y is being_line by AFF_4:11;
      reconsider A=[Y,1] as LINE of IncProjSp_of(AS) by A20,Th23;
      take A;
      thus a on A & b on A by A18,A19,A20,Th26;
    end;
    hence thesis by A14,Th20;
  end;
  hence thesis by A1,Th20;
end;

Lm5: AS is AffinPlane implies ex a st a on A & a on K
proof
  consider X such that
A1: A=[X,1] & X is being_line or A=[PDir(X),2] & X is being_plane by Th23;
  consider X9 such that
A2: K=[X9,1] & X9 is being_line or K=[PDir(X9),2] & X9 is being_plane by Th23;
  assume
A3: AS is AffinPlane;
A4: now
    assume that
A5: A=[X,1] and
A6: X is being_line;
A7: now
      assume that
A8:   K=[X9,1] and
A9:   X9 is being_line;
A10:  now
        reconsider a=LDir(X),b=LDir(X9) as Element of the Points of
        IncProjSp_of(AS) by A6,A9,Th20;
        X9 // X9 by A9,AFF_1:41;
        then
A11:    X9 '||' X9 by A9,AFF_4:40;
        assume X // X9;
        then
A12:    a=b by A6,A9,Th11;
        take a;
        X // X by A6,AFF_1:41;
        then X '||' X by A6,AFF_4:40;
        hence a on A & a on K by A5,A6,A8,A9,A12,A11,Th28;
      end;
      now
        assume not X // X9;
        then consider x such that
A13:    x in X and
A14:    x in X9 by A3,A6,A9,AFF_1:58;
        reconsider a=x as Element of the Points of IncProjSp_of(AS) by Th20;
        take a;
        thus a on A & a on K by A5,A6,A8,A9,A13,A14,Th26;
      end;
      hence thesis by A10;
    end;
    now
      X // X by A6,AFF_1:41;
      then
A15:  X '||' X by A6,AFF_4:40;
      reconsider a=LDir(X) as Element of the Points of IncProjSp_of(AS) by A6
,Th20;
      assume that
A16:  K=[PDir(X9),2] and
A17:  X9 is being_plane;
      take a;
      X9=the carrier of AS by A3,A17,Th2;
      then X '||' X9 by A6,A17,AFF_4:42;
      hence a on A & a on K by A5,A6,A16,A17,A15,Th28,Th29;
    end;
    hence thesis by A2,A7;
  end;
  now
    assume that
A18: A=[PDir(X),2] and
A19: X is being_plane;
A20: X=the carrier of AS by A3,A19,Th2;
A21: now
      assume that
A22:  K=[X9,1] and
A23:  X9 is being_line;
      X9 // X9 by A23,AFF_1:41;
      then
A24:  X9 '||' X9 by A23,AFF_4:40;
      reconsider a=LDir(X9) as POINT of IncProjSp_of(AS) by A23,Th20;
      take a;
      X9 '||' X by A19,A20,A23,AFF_4:42;
      hence a on A & a on K by A18,A19,A22,A23,A24,Th28,Th29;
    end;
    now
      consider a,b,c such that
A25:  a on A and
      b on A and
      c on A and
      a<>b and
      b<>c and
      c <>a by Lm3;
      assume that
A26:  K=[PDir(X9),2] and
A27:  X9 is being_plane;
      take a;
      thus a on A & a on K by A3,A18,A19,A26,A27,A25,Th3;
    end;
    hence thesis by A2,A21;
  end;
  hence thesis by A1,A4;
end;

Lm6: ex a,A st not a on A
proof
  consider x,y,z such that
A1: not LIN x,y,z by AFF_1:12;
  y<>z by A1,AFF_1:7;
  then
A2: Line(y,z) is being_line by AFF_1:def 3;
  then reconsider A=[Line(y,z),1] as LINE of IncProjSp_of(AS) by Th23;
  reconsider a=x as POINT of IncProjSp_of(AS) by Th20;
  take a,A;
  thus not a on A
  proof
    assume not thesis;
    then
A3: x in Line(y,z) by Th26;
A4: z in Line(y,z) by AFF_1:15;
    y in Line(y,z) by AFF_1:15;
    hence contradiction by A1,A2,A3,A4,AFF_1:21;
  end;
end;

theorem Th41:
  for x,y being Element of the Points of ProjHorizon(AS), X being
  Element of the Lines of IncProjSp_of(AS) st x<>y & [x,X] in the Inc of
  IncProjSp_of(AS) & [y,X] in the Inc of IncProjSp_of(AS) ex Y being Element of
  the Lines of ProjHorizon(AS) st X=[Y,2]
proof
  let x,y be Element of the Points of ProjHorizon(AS), X be Element of the
  Lines of IncProjSp_of(AS);
  reconsider a=x,b=y as POINT of IncProjSp_of(AS) by Th22;
  assume that
A1: x<>y and
A2: [x,X] in the Inc of IncProjSp_of(AS) and
A3: [y,X] in the Inc of IncProjSp_of(AS);
A4: b on X by A3,INCSP_1:def 1;
  consider Y being Element of the Lines of ProjHorizon(AS) such that
A5: x on Y and
A6: y on Y by Th40;
  reconsider A=[Y,2] as LINE of IncProjSp_of(AS) by Th25;
  consider Z being Subset of AS such that
A7: Y=PDir(Z) and
A8: Z is being_plane by Th15;
  consider X2 being Subset of AS such that
A9: y=LDir(X2) and
A10: X2 is being_line by Th14;
  X2 '||' Z by A9,A10,A6,A7,A8,Th36;
  then
A11: b on A by A9,A10,A7,A8,Th29;
  take Y;
  consider X1 being Subset of AS such that
A12: x=LDir(X1) and
A13: X1 is being_line by Th14;
  X1 '||' Z by A12,A13,A5,A7,A8,Th36;
  then
A14: a on A by A12,A13,A7,A8,Th29;
  a on X by A2,INCSP_1:def 1;
  hence thesis by A1,A4,A14,A11,Lm2;
end;

theorem Th42:
  for x being POINT of IncProjSp_of(AS),X being Element of the
Lines of ProjHorizon(AS) st [x,[X,2]] in the Inc of IncProjSp_of(AS) holds x is
  Element of the Points of ProjHorizon(AS)
proof
  let x be POINT of IncProjSp_of(AS), X be Element of the Lines of ProjHorizon
  (AS) such that
A1: [x,[X,2]] in the Inc of IncProjSp_of(AS);
  reconsider A=[X,2] as LINE of IncProjSp_of(AS) by Th25;
A2: ex Y st X=PDir(Y) & Y is being_plane by Th15;
  not x is Element of AS
  proof
    assume not thesis;
    then reconsider a=x as Element of AS;
A3: a=x;
    x on A by A1,INCSP_1:def 1;
    hence contradiction by A2,A3,Th27;
  end;
  then ex Y9 st x=LDir(Y9) & Y9 is being_line by Th20;
  hence thesis by Th14;
end;

Lm7: X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y
& A=[X,1] & K=[X9,1] & b on A & c on K & b on M & c on M & b<>c & M=[Y9,1] & Y9
is being_line implies Y9 c= Y
proof
  assume that
A1: X is being_line and
A2: X9 is being_line and
A3: Y is being_plane and
A4: X c= Y and
A5: X9 c= Y and
A6: A=[X,1] and
A7: K=[X9,1] and
A8: b on A and
A9: c on K and
A10: b on M and
A11: c on M and
A12: b<>c and
A13: M=[Y9,1] and
A14: Y9 is being_line;
A15: now
    assume b is Element of AS;
    then reconsider y=b as Element of AS;
A16: now
      given Xc being Subset of AS such that
A17:  c =LDir(Xc) and
A18:  Xc is being_line;
      Xc '||' X9 by A2,A7,A9,A17,A18,Th28;
      then
A19:  Xc // X9 by A2,A18,AFF_4:40;
      Xc '||' Y9 by A11,A13,A14,A17,A18,Th28;
      then Xc // Y9 by A14,A18,AFF_4:40;
      then
A20:  X9 // Y9 by A19,AFF_1:44;
      y in Y9 by A10,A13,Th26;
      then
A21:  Y9= y*X9 by A2,A20,AFF_4:def 3;
      y in X by A6,A8,Th26;
      hence thesis by A2,A3,A4,A5,A21,AFF_4:28;
    end;
    now
      assume c is Element of AS;
      then reconsider z=c as Element of AS;
A22:  z in Y9 by A11,A13,Th26;
      y in Y9 by A10,A13,Th26;
      then
A23:  Y9=Line(y,z) by A12,A14,A22,AFF_1:57;
A24:  z in X9 by A7,A9,Th26;
      y in X by A6,A8,Th26;
      hence thesis by A3,A4,A5,A12,A24,A23,AFF_4:19;
    end;
    hence thesis by A16,Th20;
  end;
  now
    given Xb being Subset of AS such that
A25: b=LDir(Xb) and
A26: Xb is being_line;
A27: now
      assume c is Element of AS;
      then reconsider y=c as Element of AS;
      Xb '||' X by A1,A6,A8,A25,A26,Th28;
      then
A28:  Xb // X by A1,A26,AFF_4:40;
      Xb '||' Y9 by A10,A13,A14,A25,A26,Th28;
      then Xb // Y9 by A14,A26,AFF_4:40;
      then
A29:  X // Y9 by A28,AFF_1:44;
      y in Y9 by A11,A13,Th26;
      then
A30:  Y9=y*X by A1,A29,AFF_4:def 3;
      y in X9 by A7,A9,Th26;
      hence thesis by A1,A3,A4,A5,A30,AFF_4:28;
    end;
    now
      Xb '||' Y9 by A10,A13,A14,A25,A26,Th28;
      then
A31:  Xb // Y9 by A14,A26,AFF_4:40;
      given Xc being Subset of AS such that
A32:  c =LDir(Xc) and
A33:  Xc is being_line;
      Xc '||' Y9 by A11,A13,A14,A32,A33,Th28;
      then Xc // Y9 by A14,A33,AFF_4:40;
      then Xc // Xb by A31,AFF_1:44;
      hence contradiction by A12,A25,A26,A32,A33,Th11;
    end;
    hence thesis by A27,Th20;
  end;
  hence thesis by A15,Th20;
end;

Lm8: X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y
& A=[X,1] & K=[X9,1] & b on A & c on K & b on M & c on M & b<>c & M=[PDir(Y9),2
] & Y9 is being_plane implies Y9 '||' Y & Y '||' Y9
proof
  assume that
A1: X is being_line and
A2: X9 is being_line and
A3: Y is being_plane and
A4: X c= Y and
A5: X9 c= Y and
A6: A=[X,1] and
A7: K=[X9,1] and
A8: b on A and
A9: c on K and
A10: b on M and
A11: c on M and
A12: b<>c and
A13: M=[PDir(Y9),2] and
A14: Y9 is being_plane;
  b is Element of AS or ex Xb being Subset of AS st b=LDir(Xb) & Xb is
  being_line by Th20;
  then consider Xb being Subset of AS such that
A15: b=LDir(Xb) and
A16: Xb is being_line by A10,A13,Th27;
A17: Xb '||' Y9 by A10,A13,A14,A15,A16,Th29;
  Xb '||' X by A1,A6,A8,A15,A16,Th28;
  then Xb // X by A1,A16,AFF_4:40;
  then
A18: Xb '||' Y by A1,A3,A4,AFF_4:42,56;
  c is Element of AS or ex Xc being Subset of AS st c =LDir(Xc) & Xc is
  being_line by Th20;
  then consider Xc being Subset of AS such that
A19: c =LDir(Xc) and
A20: Xc is being_line by A11,A13,Th27;
A21: Xc '||' Y9 by A11,A13,A14,A19,A20,Th29;
  Xc '||' X9 by A2,A7,A9,A19,A20,Th28;
  then Xc // X9 by A2,A20,AFF_4:40;
  then
A22: Xc '||' Y by A2,A3,A5,AFF_4:42,56;
  not Xb // Xc by A12,A15,A16,A19,A20,Th11;
  hence thesis by A3,A14,A16,A20,A17,A21,A18,A22,Th5;
end;

theorem Th43:
  Y is being_plane & X is being_line & X9 is being_line & X c= Y &
  X9 c= Y & P=[X,1] & Q=[X9,1] implies ex q st q on P & q on Q
proof
  assume that
A1: Y is being_plane and
A2: X is being_line and
A3: X9 is being_line and
A4: X c= Y and
A5: X9 c= Y and
A6: P=[X,1] and
A7: Q=[X9,1];
A8: now
    reconsider q=LDir(X) as POINT of IncProjSp_of(AS) by A2,Th20;
    assume
A9: X // X9;
    take q;
    LDir(X)=LDir(X9) by A2,A3,A9,Th11;
    hence q on P & q on Q by A2,A3,A6,A7,Th30;
  end;
  now
    given y such that
A10: y in X and
A11: y in X9;
    reconsider q=y as Element of the Points of IncProjSp_of(AS) by Th20;
    take q;
    thus q on P & q on Q by A2,A3,A6,A7,A10,A11,Th26;
  end;
  hence thesis by A1,A2,A3,A4,A5,A8,AFF_4:22;
end;

Lm9: a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b
on Q & d on Q & not p on P & not p on Q & M<>N & p is Element of AS implies ex
q st q on P & q on Q
proof
  assume that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N and
A14: p is Element of AS;
A15: b<>d by A2,A4,A5,A6,A9,A12,A13,Lm2;
  reconsider x=p as Element of AS by A14;
  consider XM being Subset of AS such that
A16: M=[XM,1] & XM is being_line or M=[PDir(XM),2] & XM is being_plane by Th23;
  consider XQ being Subset of AS such that
A17: Q=[XQ,1] & XQ is being_line or Q=[PDir(XQ),2] & XQ is being_plane by Th23;
  consider XP being Subset of AS such that
A18: P=[XP,1] & XP is being_line or P=[PDir(XP),2] & XP is being_plane by Th23;
  consider XN being Subset of AS such that
A19: N=[XN,1] & XN is being_line or N=[PDir(XN),2] & XN is being_plane by Th23;
A20: x in XM by A5,A16,Th26,Th27;
  x in XN by A6,A19,Th26,Th27;
  then consider Y such that
A21: XM c= Y and
A22: XN c= Y and
A23: Y is being_plane by A5,A6,A16,A19,A20,Th27,AFF_4:38;
A24: a<>c by A1,A3,A5,A6,A7,A11,A13,Lm2;
A25: now
    assume that
A26: P=[PDir(XP),2] and
A27: XP is being_plane;
A28: Y '||' XP by A1,A3,A5,A6,A7,A8,A24,A16,A19,A20,A21,A22,A23,A26,A27,Lm8
,Th27;
A29: now
      assume that
A30:  Q=[XQ,1] and
A31:  XQ is being_line;
      reconsider q=LDir(XQ) as POINT of IncProjSp_of(AS) by A31,Th20;
      take q;
      XQ c= Y by A2,A4,A5,A6,A9,A10,A15,A16,A19,A20,A21,A22,A23,A30,A31,Lm7
,Th27;
      then XQ '||' Y by A23,A31,AFF_4:42;
      then XQ '||' XP by A23,A28,AFF_4:59,60;
      hence q on P by A26,A27,A31,Th29;
      thus q on Q by A30,A31,Th30;
    end;
    now
      consider q,r,p9 such that
A32:  q on P and
      r on P and
      p9 on P and
      q<>r and
      r<>p9 and
      p9<>q by Lm3;
      assume that
A33:  Q=[PDir(XQ),2] and
A34:  XQ is being_plane;
      take q;
      thus q on P by A32;
      Y '||' XQ by A2,A4,A5,A6,A9,A10,A15,A16,A19,A20,A21,A22,A23,A33,A34,Lm8
,Th27;
      then XP '||' XQ by A23,A27,A28,A34,AFF_4:61;
      hence q on Q by A26,A27,A33,A34,A32,Th13;
    end;
    hence thesis by A17,A29;
  end;
  now
    assume that
A35: P=[XP,1] and
A36: XP is being_line;
A37: XP c= Y by A1,A3,A5,A6,A7,A8,A24,A16,A19,A20,A21,A22,A23,A35,A36,Lm7,Th27;
A38: now
A39:  XP '||' Y by A23,A36,A37,AFF_4:42;
      reconsider q=LDir(XP) as POINT of IncProjSp_of(AS) by A36,Th20;
      assume that
A40:  Q=[PDir(XQ),2] and
A41:  XQ is being_plane;
      take q;
      thus q on P by A35,A36,Th30;
      Y '||' XQ by A2,A4,A5,A6,A9,A10,A15,A16,A19,A20,A21,A22,A23,A40,A41,Lm8
,Th27;
      then XP '||' XQ by A23,A39,AFF_4:59,60;
      hence q on Q by A36,A40,A41,Th29;
    end;
    now
      assume that
A42:  Q=[XQ,1] and
A43:  XQ is being_line;
      XQ c= Y by A2,A4,A5,A6,A9,A10,A15,A16,A19,A20,A21,A22,A23,A42,A43,Lm7
,Th27;
      hence thesis by A23,A35,A36,A37,A42,A43,Th43;
    end;
    hence thesis by A17,A38;
  end;
  hence thesis by A18,A25;
end;

Lm10: a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P &
b on Q & d on Q & not p on P & not p on Q & M<>N & not p is Element of AS & a
is Element of AS implies ex q st q on P & q on Q
proof
  assume that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N and
A14: not p is Element of AS and
A15: a is Element of AS;
  reconsider x=a as Element of AS by A15;
  consider XM being Subset of AS such that
A16: M=[XM,1] & XM is being_line or M=[PDir(XM),2] & XM is being_plane by Th23;
A17: x in XM by A1,A16,Th26,Th27;
A18: b<>d by A2,A4,A5,A6,A9,A12,A13,Lm2;
  consider XN being Subset of AS such that
A19: N=[XN,1] & XN is being_line or N=[PDir(XN),2] & XN is being_plane by Th23;
  consider XP being Subset of AS such that
A20: P=[XP,1] & XP is being_line or P=[PDir(XP),2] & XP is being_plane by Th23;
A21: x=a;
  then reconsider y=b as Element of AS by A1,A2,A5,A9,A12,A14,A16,Th27,Th35;
A22: y in XM by A2,A16,Th26,Th27;
  consider X such that
A23: p=LDir(X) and
A24: X is being_line by A14,Th20;
  consider XQ being Subset of AS such that
A25: Q=[XQ,1] & XQ is being_line or Q=[PDir(XQ),2] & XQ is being_plane by Th23;
A26: x in XP by A7,A20,Th26,Th27;
  then consider Y such that
A27: XM c= Y and
A28: XP c= Y and
A29: Y is being_plane by A1,A7,A16,A20,A17,Th27,AFF_4:38;
A30: y=b;
A31: X '||' XM by A1,A5,A23,A24,A16,A21,Th27,Th28;
  then
A32: X // XM by A1,A24,A16,A21,Th27,AFF_4:40;
A33: y in XQ by A9,A25,Th26,Th27;
A34: not XM // XP by A1,A5,A7,A11,A16,A20,A17,A26,Th27,AFF_1:45;
A35: now
A36: X // XM by A1,A24,A16,A21,A31,Th27,AFF_4:40;
    assume that
A37: N=[XN,1] and
A38: XN is being_line;
    X '||' XN by A6,A23,A24,A37,A38,Th28;
    then X // XN by A24,A38,AFF_4:40;
    then
A39: XM // XN by A36,AFF_1:44;
    c is Element of AS
    proof
      assume not thesis;
      then c =LDir( XN ) by A3,A37,A38,Th34;
      then XN '||' XP by A7,A8,A20,A21,A38,Th27,Th28;
      then XN // XP by A7,A20,A21,A38,Th27,AFF_4:40;
      hence contradiction by A34,A39,AFF_1:44;
    end;
    then reconsider z=c as Element of AS;
    z in XN by A3,A37,Th26;
    then
A40: XN=z*XM by A1,A16,A21,A39,Th27,AFF_4:def 3;
A41: not XN // XQ
    proof
      assume XN // XQ;
      then XM // XQ by A39,AFF_1:44;
      hence contradiction by A2,A5,A9,A12,A16,A25,A33,A22,Th27,AFF_1:45;
    end;
    now
      assume not d is Element of AS;
      then consider Xd being Subset of AS such that
A42:  d=LDir(Xd) and
A43:  Xd is being_line by Th20;
      Xd '||' XN by A4,A37,A38,A42,A43,Th28;
      then
A44:  Xd // XN by A38,A43,AFF_4:40;
      Xd '||' XQ by A9,A10,A25,A30,A42,A43,Th27,Th28;
      then Xd // XQ by A9,A25,A30,A43,Th27,AFF_4:40;
      hence contradiction by A41,A44,AFF_1:44;
    end;
    then reconsider w=d as Element of AS;
    w in XQ by A10,A25,Th26,Th27;
    then
A45: XQ=Line(y,w) by A9,A18,A25,A33,Th27,AFF_1:57;
    z in XP by A8,A20,Th26,Th27;
    then
A46: XN c= Y by A1,A16,A21,A27,A28,A29,A40,Th27,AFF_4:28;
    w in XN by A4,A37,Th26;
    then XQ c= Y by A18,A27,A29,A22,A46,A45,AFF_4:19;
    hence thesis by A7,A9,A20,A25,A21,A28,A29,A30,Th27,Th43;
  end;
A47: XP '||' Y by A7,A20,A21,A28,A29,Th27,AFF_4:42;
A48: XM '||' Y by A1,A16,A21,A27,A29,Th27,AFF_4:42;
  now
    assume that
A49: N=[PDir(XN),2] and
A50: XN is being_plane;
    c is not Element of AS by A3,A49,Th27;
    then c =LDir(XP) by A7,A8,A20,A21,Th27,Th34;
    then
A51: XP '||' XN by A3,A7,A20,A21,A49,A50,Th27,Th29;
    d is not Element of AS by A4,A49,Th27;
    then d=LDir(XQ) by A9,A10,A25,A30,Th27,Th34;
    then
A52: XQ '||' XN by A4,A9,A25,A30,A49,A50,Th27,Th29;
    X '||' XN by A6,A23,A24,A49,A50,Th29;
    then XM '||' XN by A32,AFF_4:56;
    then XN '||' Y by A1,A5,A7,A11,A16,A20,A17,A26,A29,A48,A47,A50,A51,Th5,Th27
,AFF_1:45;
    then XQ '||' Y by A50,A52,AFF_4:59,60;
    then XQ c= Y by A9,A25,A27,A29,A33,A22,Th27,AFF_4:43;
    hence thesis by A7,A9,A20,A25,A21,A28,A29,A30,Th27,Th43;
  end;
  hence thesis by A19,A35;
end;

Lm11: a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P &
b on Q & d on Q & not p on P & not p on Q & M<>N & not p is Element of AS & (a
is Element of AS or b is Element of AS or c is Element of AS or d is Element of
AS) implies ex q st q on P & q on Q
proof
  assume that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N and
A14: not p is Element of AS and
A15: a is Element of AS or b is Element of AS or c is Element of AS or d
  is Element of AS;
  now
    assume b is Element of AS or d is Element of AS;
    then
    ex q st q on Q & q on P by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14
,Lm10;
    hence thesis;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,Lm10;
end;

Lm12: a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P &
b on Q & d on Q & not p on P & not p on Q & M<>N implies ex q st q on P & q on
Q
proof
  assume that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N;
  now
    assume
A14: not p is Element of AS;
    now
A15:  b<>d by A2,A4,A5,A6,A9,A12,A13,Lm2;
      set x = the Element of AS;
      assume that
A16:  not a is Element of AS and
A17:  not b is Element of AS and
A18:  not c is Element of AS and
A19:  not d is Element of AS;
      consider Xa being Subset of AS such that
A20:  a=LDir(Xa) and
A21:  Xa is being_line by A16,Th20;
      consider Xc being Subset of AS such that
A22:  c =LDir(Xc) and
A23:  Xc is being_line by A18,Th20;
      consider Xb being Subset of AS such that
A24:  b=LDir(Xb) and
A25:  Xb is being_line by A17,Th20;
      consider Xd being Subset of AS such that
A26:  d=LDir(Xd) and
A27:  Xd is being_line by A19,Th20;
      consider Xp being Subset of AS such that
A28:  p=LDir(Xp) and
A29:  Xp is being_line by A14,Th20;
      set Xa9=x*Xa,Xb9=x*Xb,Xc9=x*Xc,Xd9=x*Xd,Xp9=x*Xp;
      consider y such that
A30:  x<>y and
A31:  y in Xa9 by A21,AFF_1:20,AFF_4:27;
A32:  Xp // Xp9 by A29,AFF_4:def 3;
      consider t such that
A33:  x<>t and
A34:  t in Xc9 by A23,AFF_1:20,AFF_4:27;
      set Y1=y*Xp9,Y2=t*Xp9;
A35:  Xp9 is being_line by A29,AFF_4:27;
      then
A36:  Y1 is being_line by AFF_4:27;
A37:  Y2 is being_line by A35,AFF_4:27;
A38:  Xb // Xb9 by A25,AFF_4:def 3;
A39:  Xd9 is being_line by A27,AFF_4:27;
A40:  Xd // Xd9 by A27,AFF_4:def 3;
A41:  x in Xc9 by A23,AFF_4:def 3;
A42:  x in Xb9 by A25,AFF_4:def 3;
A43:  Xb9 is being_line by A25,AFF_4:27;
A44:  x in Xd9 by A27,AFF_4:def 3;
      then consider XQ being Subset of AS such that
A45:  Xb9 c= XQ and
A46:  Xd9 c= XQ and
A47:  XQ is being_plane by A43,A39,A42,AFF_4:38;
A48:  Xa9 is being_line by A21,AFF_4:27;
A49:  Xp9 // Y2 by A35,AFF_4:def 3;
A50:  not Xd9 // Y2
      proof
        assume Xd9 // Y2;
        then Xd // Y2 by A40,AFF_1:44;
        then Xd // Xp9 by A49,AFF_1:44;
        then Xd // Xp by A32,AFF_1:44;
        hence contradiction by A10,A12,A28,A29,A26,A27,Th11;
      end;
A51:  Xp9 // Y1 by A35,AFF_4:def 3;
A52:  not Xb9 // Y1
      proof
        assume Xb9 // Y1;
        then Xb // Y1 by A38,AFF_1:44;
        then Xb // Xp9 by A51,AFF_1:44;
        then Xb // Xp by A32,AFF_1:44;
        hence contradiction by A9,A12,A28,A29,A24,A25,Th11;
      end;
A53:  x in Xa9 by A21,AFF_4:def 3;
A54:  Xc9 is being_line by A23,AFF_4:27;
      then consider XP being Subset of AS such that
A55:  Xa9 c= XP and
A56:  Xc9 c= XP and
A57:  XP is being_plane by A48,A53,A41,AFF_4:38;
A58:  x in Xp9 by A29,AFF_4:def 3;
      then consider X2 being Subset of AS such that
A59:  Xc9 c= X2 and
A60:  Xp9 c= X2 and
A61:  X2 is being_plane by A54,A35,A41,AFF_4:38;
A62:  Xc // Xc9 by A23,AFF_4:def 3;
      N=[PDir(X2),2]
      proof
        reconsider N9=[PDir(X2),2] as Element of the Lines of IncProjSp_of(AS)
        by A61,Th23;
A63:    c on N9 by A22,A62,A59,A61,Th32;
        p on N9 by A28,A32,A60,A61,Th32;
        hence thesis by A3,A6,A8,A11,A63,Lm2;
      end;
      then Xd '||' X2 by A4,A26,A27,A61,Th29;
      then
A64:  Xd9 c= X2 by A39,A41,A44,A40,A59,A61,AFF_4:43,56;
      consider X1 being Subset of the carrier of AS such that
A65:  Xa9 c= X1 and
A66:  Xp9 c= X1 and
A67:  X1 is being_plane by A48,A35,A53,A58,AFF_4:38;
A68:  Xa // Xa9 by A21,AFF_4:def 3;
      M=[PDir(X1),2]
      proof
        reconsider M9=[PDir(X1),2] as Element of the Lines of IncProjSp_of(AS)
        by A67,Th23;
A69:    a on M9 by A20,A68,A65,A67,Th32;
        p on M9 by A28,A32,A66,A67,Th32;
        hence thesis by A1,A5,A7,A11,A69,Lm2;
      end;
      then Xb '||' X1 by A2,A24,A25,A67,Th29;
      then
A70:  Xb9 c= X1 by A43,A53,A42,A38,A65,A67,AFF_4:43,56;
      Y1 c= X1 by A29,A31,A65,A66,A67,AFF_4:27,28;
      then consider z such that
A71:  z in Xb9 and
A72:  z in Y1 by A43,A36,A67,A70,A52,AFF_4:22;
      Y2 c= X2 by A29,A34,A59,A60,A61,AFF_4:27,28;
      then consider u such that
A73:  u in Xd9 and
A74:  u in Y2 by A39,A37,A61,A64,A50,AFF_4:22;
      set AC=Line(y,t),BD=Line(z,u);
A75:  y in AC by AFF_1:15;
A76:  y in Y1 by A35,AFF_4:def 3;
A77:  x<>z
      proof
        assume
A78:    not thesis;
        a = LDir(Xa9) by A20,A21,A48,A68,Th11
          .= LDir(Y1) by A48,A53,A30,A31,A36,A76,A72,A78,AFF_1:18
          .= LDir(Xp9) by A35,A36,A51,Th11
          .= p by A28,A29,A35,A32,Th11;
        hence contradiction by A7,A11;
      end;
A79:  z<>u
      proof
        assume
A80:    not thesis;
        b= LDir(Xb9) by A24,A25,A43,A38,Th11
          .= LDir(Xd9) by A43,A39,A42,A44,A71,A73,A77,A80,AFF_1:18
          .= d by A26,A27,A39,A40,Th11;
        hence contradiction by A2,A4,A5,A6,A9,A12,A13,Lm2;
      end;
      then
A81:  BD is being_line by AFF_1:def 3;
A82:  Xa9<>Xc9
      proof
        assume Xa9=Xc9;
        then a=LDir(Xc9) by A20,A21,A48,A68,Th11
          .= c by A22,A23,A54,A62,Th11;
        hence contradiction by A1,A3,A5,A6,A7,A11,A13,Lm2;
      end;
      then
A83:  y<>t by A48,A54,A53,A41,A30,A31,A34,AFF_1:18;
      then
A84:  AC is being_line by AFF_1:def 3;
A85:  BD c= XQ by A71,A73,A79,A45,A46,A47,AFF_4:19;
A86:  t in AC by AFF_1:15;
      Y1 // Y2 by A51,A49,AFF_1:44;
      then consider X3 being Subset of AS such that
A87:  Y1 c= X3 and
A88:  Y2 c= X3 and
A89:  X3 is being_plane by AFF_4:39;
A90:  BD c= X3 by A87,A88,A89,A72,A74,A79,AFF_4:19;
A91:  a<>c by A1,A3,A5,A6,A7,A11,A13,Lm2;
A92:  P=[PDir(XP),2] & Q=[PDir(XQ),2]
      proof
        reconsider P9=[PDir(XP),2], Q9=[PDir(XQ),2] as LINE of IncProjSp_of(AS
        ) by A57,A47,Th23;
A93:    c on P9 by A22,A62,A56,A57,Th32;
A94:    b on Q9 by A24,A38,A45,A47,Th32;
A95:    d on Q9 by A26,A40,A46,A47,Th32;
        a on P9 by A20,A68,A55,A57,Th32;
        hence thesis by A7,A8,A9,A10,A91,A15,A93,A94,A95,Lm2;
      end;
A96:  now
        reconsider q=LDir(AC),q9=LDir(BD) as Element of the Points of
        IncProjSp_of(AS) by A84,A81,Th20;
        assume
A97:    AC // BD;
        take q;
        q=q9 by A84,A81,A97,Th11;
        hence q on P & q on Q by A31,A34,A71,A73,A83,A79,A84,A81,A55,A56,A57
,A45,A46,A47,A92,Th31,AFF_4:19;
      end;
A98:  AC c= XP by A31,A34,A83,A55,A56,A57,AFF_4:19;
A99:  now
        given w such that
A100:   w in AC and
A101:   w in BD;
        set R=Line(x,w);
A102:   x<>w
        proof
          assume
A103:     x=w;
          then Xa9=AC by A48,A53,A30,A31,A84,A75,A100,AFF_1:18;
          hence contradiction by A54,A41,A33,A34,A82,A84,A86,A100,A103,AFF_1:18
;
        end;
        then
A104:   R is being_line by AFF_1:def 3;
        then reconsider q=LDir(R) as POINT of IncProjSp_of(AS) by Th20;
        take q;
        thus q on P & q on Q by A53,A42,A55,A57,A45,A47,A92,A98,A85,A100,A101
,A102,A104,Th31,AFF_4:19;
      end;
      t in Y2 by A35,AFF_4:def 3;
      then AC c= X3 by A76,A87,A88,A89,A83,AFF_4:19;
      hence thesis by A89,A84,A81,A90,A96,A99,AFF_4:22;
    end;
    hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,Lm11;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,Lm9;
end;

theorem Th44:
  for a,b,c,d,p being Element of the Points of ProjHorizon(AS),M,N
,P,Q being Element of the Lines of ProjHorizon(AS) st a on M & b on M & c on N
  & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P &
not p on Q & M<>N ex q being Element of the Points of ProjHorizon(AS) st q on P
  & q on Q
proof
  let a,b,c,d,p be Element of the Points of ProjHorizon(AS),M,N,P,Q be Element
  of the Lines of ProjHorizon(AS) such that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N;
  reconsider M9=[M,2],N9=[N,2],P9=[P,2],Q9=[Q,2] as LINE of IncProjSp_of(AS)
  by Th25;
  reconsider a9=a,b9=b,c9=c,d9=d,p9=p as POINT of IncProjSp_of(AS) by Th22;
A14: b9 on M9 by A2,Th37;
A15: M9<>N9
  proof
    assume M9=N9;
    then M = [N,2]`1
      .= N;
    hence contradiction by A13;
  end;
A16: d9 on N9 by A4,Th37;
A17: c9 on N9 by A3,Th37;
A18: c9 on P9 by A8,Th37;
A19: a9 on P9 by A7,Th37;
A20: p9 on N9 by A6,Th37;
A21: p9 on M9 by A5,Th37;
A22: not p9 on Q9 by A12,Th37;
A23: not p9 on P9 by A11,Th37;
A24: d9 on Q9 by A10,Th37;
A25: b9 on Q9 by A9,Th37;
  a9 on M9 by A1,Th37;
  then consider q9 being POINT of IncProjSp_of(AS) such that
A26: q9 on P9 and
A27: q9 on Q9 by A14,A17,A16,A21,A20,A19,A18,A25,A24,A23,A22,A15,Lm12;
  [q9,[P,2]] in the Inc of IncProjSp_of(AS) by A26,INCSP_1:def 1;
  then reconsider q=q9 as Element of the Points of ProjHorizon(AS) by Th42;
  take q;
  thus thesis by A26,A27,Th37;
end;

registration
  let AS;
  cluster IncProjSp_of(AS) -> partial linear up-2-dimensional up-3-rank
    Vebleian;
  correctness
  proof
    set XX = IncProjSp_of(AS);
A1: for a,b being POINT of XX ex A being LINE of XX st a on A & b on A by Lm4;
A2: ex a being POINT of XX, A being LINE of XX st not a on A by Lm6;
A3: for A being LINE of XX ex a,b,c being POINT of XX st a<>b & b<>c & c
    <>a & a on A & b on A & c on A
    proof
      let A be LINE of XX;
      consider a,b,c being POINT of XX such that
A4:   a on A and
A5:   b on A and
A6:   c on A and
A7:   a<>b and
A8:   b<>c and
A9:   c <>a by Lm3;
      take a,b,c;
      thus thesis by A4,A5,A6,A7,A8,A9;
    end;
A10: for a,b,c,d,p,q being POINT of XX, M,N,P,Q being LINE of XX st a on M
& b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on
Q & not p on P & not p on Q & M<>N holds ex q being POINT of XX st q on P & q
    on Q by Lm12;
    for a,b being POINT of XX, A,K being LINE of XX st a on A & b on A & a
    on K & b on K holds a=b or A=K by Lm2;
    hence thesis by A1,A2,A3,A10,INCPROJ:def 4,def 5,def 6,def 7,def 8;
  end;
end;

registration
  cluster strict 2-dimensional for IncProjSp;
  existence
  proof
    set AS = the AffinPlane;
    set XX=IncProjSp_of(AS);
    for A,K being LINE of XX ex a being POINT of XX st a on A & a on K by Lm5;
    then XX is 2-dimensional IncProjSp by INCPROJ:def 9;
    hence thesis;
  end;
end;

registration
  let AS be AffinPlane;
  cluster IncProjSp_of(AS) -> 2-dimensional;
  correctness
  proof
    set XX=IncProjSp_of(AS);
    for A,K being LINE of XX ex a being Element of the Points of XX st a
    on A & a on K by Lm5;
    hence thesis by INCPROJ:def 9;
  end;
end;

theorem
  IncProjSp_of(AS) is 2-dimensional implies AS is AffinPlane
proof
  set x = the Element of AS;
  assume
A1: IncProjSp_of(AS) is 2-dimensional;
  consider X such that
A2: x in X and
  x in X and
  x in X and
A3: X is being_plane by AFF_4:37;
  assume AS is not AffinPlane;
  then consider z such that
A4: not z in X by A3,AFF_4:48;
  set Y=Line(x,z);
A5: Y is being_line by A2,A4,AFF_1:def 3;
  then reconsider A=[PDir(X),2],K=[Y,1] as LINE of IncProjSp_of(AS) by A3,Th23;
  consider a being POINT of IncProjSp_of(AS) such that
A6: a on A and
A7: a on K by A1,INCPROJ:def 9;
  not a is Element of AS by A6,Th27;
  then consider Y9 such that
A8: a=LDir(Y9) and
A9: Y9 is being_line by Th20;
  Y9 '||' Y by A5,A7,A8,A9,Th28;
  then
A10: Y9 // Y by A5,A9,AFF_4:40;
A11: z in Y by AFF_1:15;
A12: x in Y by AFF_1:15;
  Y9 '||' X by A3,A6,A8,A9,Th29;
  then Y c= X by A2,A3,A5,A12,A10,AFF_4:43,56;
  hence contradiction by A4,A11;
end;

theorem
  AS is not AffinPlane implies ProjHorizon(AS) is IncProjSp
proof
  set XX = ProjHorizon(AS);
A1: for a,b being Element of the Points of XX ex A being Element of the
  Lines of XX st a on A & b on A by Th40;
A2: for A being Element of the Lines of XX ex a,b,c being Element of the
  Points of XX st a<>b & b<>c & c <>a & a on A & b on A & c on A
  proof
    let A be Element of the Lines of XX;
    consider a,b,c being Element of the Points of XX such that
A3: a on A and
A4: b on A and
A5: c on A and
A6: a<>b and
A7: b<>c and
A8: c <>a by Th39;
    take a,b,c;
    thus thesis by A3,A4,A5,A6,A7,A8;
  end;
  assume AS is not AffinPlane;
  then
A9: ex a being Element of the Points of XX, A being Element of the Lines of
  XX st not a on A by Lm1;
A10: for a,b,c,d,p,q being Element of the Points of XX, M,N,P,Q being
Element of the Lines of XX st a on M & b on M & c on N & d on N & p on M & p on
N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M<>N holds ex
  q being Element of the Points of XX st q on P & q on Q by Th44;
  for a,b being Element of the Points of XX, A,K being Element of the
  Lines of XX st a on A & b on A & a on K & b on K holds a=b or A=K by Th38;
  hence thesis by A1,A9,A2,A10,INCPROJ:def 4,def 5,def 6,def 7,def 8;
end;

theorem
  ProjHorizon(AS) is IncProjSp implies AS is not AffinPlane
proof
  set XX=ProjHorizon(AS);
  assume ProjHorizon(AS) is IncProjSp;
  then consider
  a being Element of the Points of XX, A being Element of the Lines
  of XX such that
A1: not a on A by INCPROJ:def 6;
  consider X such that
A2: a=LDir(X) and
A3: X is being_line by Th14;
  consider Y such that
A4: A=PDir(Y) and
A5: Y is being_plane by Th15;
  assume AS is AffinPlane;
  then Y = the carrier of AS by A5,Th2;
  then X '||' Y by A3,A5,AFF_4:42;
  hence contradiction by A1,A2,A3,A4,A5,Th36;
end;

theorem Th48:
  for M,N being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st M is being_line & N is being_line & M<>N & o in M & o in N
  & o<>b & o<>b9 & o<>c9 & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9
  // b,a9 & b,c9 // c,b9 & (a=b or b=c or a=c) holds a,c9 // c,a9
proof
  let M,N be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
  such that
A1: M is being_line and
A2: N is being_line and
A3: M<>N and
A4: o in M and
A5: o in N and
A6: o<>b and
A7: o<>b9 and
A8: o<>c9 and
A9: b in M and
A10: c in M and
A11: a9 in N and
A12: b9 in N and
A13: c9 in N and
A14: a,b9 // b,a9 and
A15: b,c9 // c,b9 and
A16: a=b or b=c or a=c;
A17: c <>b9 by A1,A2,A3,A4,A5,A7,A10,A12,AFF_1:18;
  now
    assume
A18: a=c;
    then b,c9 // b,a9 by A14,A15,A17,AFF_1:5;
    then a9=c9 by A1,A2,A3,A4,A5,A6,A8,A9,A11,A13,AFF_4:9;
    hence thesis by A18,AFF_1:2;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A11,A12,A13,A14,A15,A16,AFF_4:9;
end;

theorem
  IncProjSp_of(AS) is Pappian implies AS is Pappian
proof
  set XX = IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Pappian;
  for M,N being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st M is being_line & N is being_line & M<>N & o in M & o in N & o
<>a & o<>a9 & o<>b & o<>b9 & o<>c & o<>c9 & a in M & b in M & c in M & a9 in N
  & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9
  proof
    let M,N be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
    such that
A2: M is being_line and
A3: N is being_line and
A4: M<>N and
A5: o in M and
A6: o in N and
A7: o<>a and
A8: o<>a9 and
A9: o<>b and
A10: o<>b9 and
A11: o<>c and
A12: o<>c9 and
A13: a in M and
A14: b in M and
A15: c in M and
A16: a9 in N and
A17: b9 in N and
A18: c9 in N and
A19: a,b9 // b,a9 and
A20: b,c9 // c,b9;
A21: b<>c9 by A2,A3,A4,A5,A6,A9,A14,A18,AFF_1:18;
    then
A22: Line(b,c9) is being_line by AFF_1:def 3;
    c <>a9 by A2,A3,A4,A5,A6,A8,A15,A16,AFF_1:18;
    then
A23: Line(c,a9) is being_line by AFF_1:def 3;
A24: b<>a9 by A2,A3,A4,A5,A6,A8,A14,A16,AFF_1:18;
    then
A25: Line(b,a9) is being_line by AFF_1:def 3;
A26: c <>b9 by A2,A3,A4,A5,A6,A10,A15,A17,AFF_1:18;
    then
A27: Line(c,b9) is being_line by AFF_1:def 3;
    reconsider A3=[M,1],B3=[N,1] as Element of the Lines of XX by A2,A3,Th23;
    reconsider p=o,a1=a9,a2=c9,a3=b9,b1=a,b2=c,b3=b as Element of the Points
    of XX by Th20;
A28: p on A3 by A2,A5,Th26;
A29: a<>b9 by A2,A3,A4,A5,A6,A7,A13,A17,AFF_1:18;
    then
A30: Line(a,b9) is being_line by AFF_1:def 3;
    then reconsider c1=LDir(Line(b,c9)),c2=LDir(Line(a,b9)) as Element of the
    Points of XX by A22,Th20;
A31: b1 on A3 by A2,A13,Th26;
    a<>c9 by A2,A3,A4,A5,A6,A7,A13,A18,AFF_1:18;
    then
A32: Line(a,c9) is being_line by AFF_1:def 3;
    then reconsider
    A1=[Line(b,c9),1],A2=[Line(b,a9),1],B1=[Line(a,b9),1], B2=[Line
(c,b9),1],C1=[Line(c,a9),1],C2=[Line(a,c9),1] as Element of the Lines of XX by
A30,A25,A22,A27,A23,Th23;
A33: c2 on B1 by A30,Th30;
A34: b3 on A3 by A2,A14,Th26;
A35: b2 on A3 by A2,A15,Th26;
    consider Y such that
A36: M c= Y and
A37: N c= Y and
A38: Y is being_plane by A2,A3,A5,A6,AFF_4:38;
    reconsider C39=[PDir(Y),2] as Element of the Lines of XX by A38,Th23;
A39: c1 on C39 by A14,A18,A36,A37,A38,A21,A22,Th31,AFF_4:19;
A40: c2 on C39 by A13,A17,A36,A37,A38,A29,A30,Th31,AFF_4:19;
A41: a1 on B3 by A3,A16,Th26;
A42: a3 on B3 by A3,A17,Th26;
A43: p on B3 by A3,A6,Th26;
    b9 in Line(a,b9) by AFF_1:15;
    then
A44: a3 on B1 by A30,Th26;
    a in Line(a,b9) by AFF_1:15;
    then
A45: b1 on B1 by A30,Th26;
A46: c in Line(c,a9) by AFF_1:15;
    then
A47: b2 on C1 by A23,Th26;
    Line(b,c9) // Line(c,b9) by A20,A21,A26,AFF_1:37;
    then Line(b,c9) '||' Line(c,b9) by A22,A27,AFF_4:40;
    then
A48: c1 on B2 by A22,A27,Th28;
A49: c9 in Line(a,c9) by AFF_1:15;
    then
A50: a2 on C2 by A32,Th26;
    b9 in Line(c,b9) by AFF_1:15;
    then
A51: a3 on B2 by A27,Th26;
    c in Line(c,b9) by AFF_1:15;
    then
A52: b2 on B2 by A27,Th26;
    c9 in Line(b,c9) by AFF_1:15;
    then
A53: a2 on A1 by A22,Th26;
    b in Line(b,c9) by AFF_1:15;
    then
A54: b3 on A1 by A22,Th26;
A55: a2 on B3 by A3,A18,Th26;
    Line(a,b9) // Line(b,a9) by A19,A29,A24,AFF_1:37;
    then Line(a,b9) '||' Line(b,a9) by A30,A25,AFF_4:40;
    then
A56: c2 on A2 by A30,A25,Th28;
A57: a in Line(a,c9) by AFF_1:15;
    then
A58: b1 on C2 by A32,Th26;
    a9 in Line(b,a9) by AFF_1:15;
    then
A59: a1 on A2 by A25,Th26;
    b in Line(b,a9) by AFF_1:15;
    then
A60: b3 on A2 by A25,Th26;
A61: a9 in Line(c,a9) by AFF_1:15;
    then
A62: a1 on C1 by A23,Th26;
A63: c1 on A1 by A22,Th30;
    now
A64:  A3<>B3
      proof
        assume A3=B3;
        then M=[N,1]`1
          .= N;
        hence contradiction by A4;
      end;
      not p on C1 & not p on C2
      proof
        assume p on C1 or p on C2;
        then a1 on A3 or a2 on A3 by A7,A11,A28,A31,A35,A58,A50,A47,A62,Lm2;
        hence contradiction by A8,A12,A28,A43,A41,A55,A64,INCPROJ:def 4;
      end;
      then consider c3 being Element of the Points of XX such that
A65:  c3 on C1 and
A66:  c3 on C2 by A28,A31,A35,A43,A41,A55,A58,A50,A47,A62,A64,INCPROJ:def 8;
A67:  {a2,b1,c3} on C2 by A58,A50,A66,INCSP_1:2;
A68:  {a1,b3,c2} on A2 by A60,A59,A56,INCSP_1:2;
A69:  {a3,b1,c2} on B1 by A45,A44,A33,INCSP_1:2;
      assume that
A70:  b1<>b2 and
A71:  b2<>b3 and
A72:  b3<>b1;
A73:  p,b1,b2,b3 are_mutually_distinct by A7,A9,A11,A70,A71,A72,ZFMISC_1:def 6
;
      a1<>a2 & a2<>a3 & a1<>a3
      proof
A74:    now
          assume a9=c9;
          then a,b9 // c,b9 by A19,A20,A24,AFF_1:5;
          hence contradiction by A2,A3,A4,A5,A6,A7,A10,A13,A15,A17,A70,AFF_4:9;
        end;
        assume not thesis;
        hence contradiction by A2,A3,A4,A5,A6,A7,A9,A10,A13,A14,A15,A17,A19,A20
,A71,A72,A74,AFF_4:9;
      end;
      then
A75:  p,a1,a2,a3 are_mutually_distinct by A8,A10,A12,ZFMISC_1:def 6;
A76:  {a1,a2,a3} on B3 by A41,A55,A42,INCSP_1:2;
A77:  {b1,b2,b3} on A3 by A31,A35,A34,INCSP_1:2;
A78:  {a3,b2,c1} on B2 by A51,A52,A48,INCSP_1:2;
A79:  {a2,b3,c1} on A1 by A53,A54,A63,INCSP_1:2;
A80:  p on B3 by A3,A6,Th26;
A81:  p on A3 by A2,A5,Th26;
A82:  {c1,c2} on C39 by A39,A40,INCSP_1:1;
      {a1,b2,c3} on C1 by A47,A62,A65,INCSP_1:2;
      then c3 on C39 by A1,A75,A73,A64,A81,A80,A79,A69,A68,A78,A67,A77,A76,A82,
INCPROJ:def 14;
      then not c3 is Element of AS by Th27;
      then consider Y such that
A83:  c3=LDir(Y) and
A84:  Y is being_line by Th20;
      Y '||' Line(c,a9) by A23,A65,A83,A84,Th28;
      then
A85:  Y // Line(c,a9) by A23,A84,AFF_4:40;
      Y '||' Line(a,c9) by A32,A66,A83,A84,Th28;
      then Y // Line(a,c9) by A32,A84,AFF_4:40;
      then Line(a,c9) // Line(c,a9) by A85,AFF_1:44;
      hence thesis by A57,A49,A46,A61,AFF_1:39;
    end;
    hence thesis by A2,A3,A4,A5,A6,A9,A10,A12,A14,A15,A16,A17,A18,A19,A20,Th48;
  end;
  hence thesis by AFF_2:def 2;
end;

theorem Th50:
  for A,P,C being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st o in A & o in P & o in C & o<>a & o<>b & o<>c & a in A & b
  in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is
being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9 & (o=a9 or a=a9) holds b
  ,c // b9,c9
proof
  let A,P,C be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
  such that
A1: o in A and
A2: o in P and
A3: o in C and
A4: o<>a and
A5: o<>b and
A6: o<>c and
A7: a in A and
A8: b in P and
A9: b9 in P and
A10: c in C and
A11: c9 in C and
A12: A is being_line and
A13: P is being_line and
A14: C is being_line and
A15: A<>P and
A16: A<>C and
A17: a,b // a9,b9 and
A18: a,c // a9,c9 and
A19: o=a9 or a=a9;
A20: now
    assume
A21: a=a9;
    then
A22: c =c9 by A1,A3,A4,A6,A7,A10,A11,A12,A14,A16,A18,AFF_4:9;
    b=b9 by A1,A2,A4,A5,A7,A8,A9,A12,A13,A15,A17,A21,AFF_4:9;
    hence thesis by A22,AFF_1:2;
  end;
  now
    assume
A23: o=a9;
    then
A24: o=c9 by A1,A3,A4,A6,A7,A10,A11,A12,A14,A16,A18,AFF_4:8;
    o=b9 by A1,A2,A4,A5,A7,A8,A9,A12,A13,A15,A17,A23,AFF_4:8;
    hence thesis by A24,AFF_1:3;
  end;
  hence thesis by A19,A20;
end;

theorem
  IncProjSp_of(AS) is Desarguesian implies AS is Desarguesian
proof
  set XX= IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Desarguesian;
  for A,P,C being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st o in A & o in P & o in C & o<>a & o<>b & o<>c & a in A & a9 in
A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C
  is being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9
  proof
    let A,P,C be Subset of AS, o,a,b,c,a9,b9,c9 be Element of
    AS such that
A2: o in A and
A3: o in P and
A4: o in C and
A5: o<>a and
A6: o<>b and
A7: o<>c and
A8: a in A and
A9: a9 in A and
A10: b in P and
A11: b9 in P and
A12: c in C and
A13: c9 in C and
A14: A is being_line and
A15: P is being_line and
A16: C is being_line and
A17: A<>P and
A18: A<>C and
A19: a,b // a9,b9 and
A20: a,c // a9,c9;
    now
      assume
A21:  P<>C;
      now
        reconsider p=o,a1=a,b1=a9,a2=b,b2=b9,a3=c,b3=c9 as Element of the
        Points of XX by Th20;
        reconsider C1=[A,1],C2=[P,1],C39=[C,1] as Element of the Lines of XX
        by A14,A15,A16,Th23;
        assume that
A22:    a<>a9 and
A23:    o<>a9;
A24:    o<>c9 by A2,A4,A5,A7,A8,A9,A12,A14,A16,A18,A20,A23,AFF_4:8;
A25:    a9<>c9 by A2,A4,A9,A13,A14,A16,A18,A23,AFF_1:18;
        then
A26:    Line(a9,c9) is being_line by AFF_1:def 3;
A27:    o<>b9 by A2,A3,A5,A6,A8,A9,A10,A14,A15,A17,A19,A23,AFF_4:8;
        then b9<>c9 by A3,A4,A11,A13,A15,A16,A21,AFF_1:18;
        then
A28:    Line(b9,c9) is being_line by AFF_1:def 3;
        b<>c by A3,A4,A6,A10,A12,A15,A16,A21,AFF_1:18;
        then
A29:    Line(b,c) is being_line by AFF_1:def 3;
A30:    a<>c by A2,A4,A5,A8,A12,A14,A16,A18,AFF_1:18;
        then
A31:    Line(a,c) is being_line by AFF_1:def 3;
A32:    a<>b by A2,A3,A5,A8,A10,A14,A15,A17,AFF_1:18;
        then
A33:    Line(a,b) is being_line by AFF_1:def 3;
        then reconsider s=LDir(Line(a,b)),r=LDir(Line(a,c)) as Element of the
        Points of XX by A31,Th20;
A34:    p on C2 by A3,A15,Th26;
A35:    a9<>b9 by A2,A3,A9,A11,A14,A15,A17,A23,AFF_1:18;
        then
A36:    Line(a9,b9) is being_line by AFF_1:def 3;
        then reconsider
        A1=[Line(b,c),1],A2=[Line(a,c),1],A3=[Line(a,b),1], B1=[
Line(b9,c9),1],B2=[Line(a9,c9),1],B3=[Line(a9,b9),1] as Element of the Lines of
        XX by A33,A29,A31,A28,A26,Th23;
A37:    r on A2 by A31,Th30;
A38:    c in Line(b,c) by AFF_1:15;
        then
A39:    a3 on A1 by A29,Th26;
A40:    a3 on A1 by A29,A38,Th26;
A41:    c9 in Line(a9,c9) by AFF_1:15;
        then
A42:    b3 on B2 by A26,Th26;
A43:    a9 in Line(a9,c9) by AFF_1:15;
        then
A44:    b1 on B2 by A26,Th26;
A45:    Line(a,c) // Line(a9,c9) by A20,A30,A25,AFF_1:37;
        then Line(a,c) '||' Line(a9,c9) by A31,A26,AFF_4:40;
        then r on B2 by A31,A26,Th28;
        then
A46:    {b1,r,b3} on B2 by A44,A42,INCSP_1:2;
A47:    c <>c9 by A2,A4,A5,A7,A8,A9,A12,A14,A16,A18,A20,A22,AFF_4:9;
A48:    b1 on C1 by A9,A14,Th26;
A49:    a3 on C39 by A12,A16,Th26;
A50:    b9 in Line(a9,b9) by AFF_1:15;
        then
A51:    b2 on B3 by A36,Th26;
A52:    a9 in Line(a9,b9) by AFF_1:15;
        then
A53:    b1 on B3 by A36,Th26;
A54:    Line(a,b) // Line(a9,b9) by A19,A32,A35,AFF_1:37;
        then Line(a,b) '||' Line(a9,b9) by A33,A36,AFF_4:40;
        then s on B3 by A33,A36,Th28;
        then
A55:    {b1,s,b2} on B3 by A53,A51,INCSP_1:2;
A56:    now
          assume C2=C39;
          then P=[C,1]`1
            .=C;
          hence contradiction by A21;
        end;
A57:    now
          assume C1=C39;
          then A=[C,1]`1
            .=C;
          hence contradiction by A18;
        end;
        now
          assume C1=C2;
          then A=[P,1]`1
            .=P;
          hence contradiction by A17;
        end;
        then
A58:    C1,C2,C39 are_mutually_distinct by A56,A57,ZFMISC_1:def 5;
A59:    a1 on C1 by A8,A14,Th26;
A60:    b3 on C39 by A13,A16,Th26;
A61:    p on C39 by A4,A16,Th26;
        then
A62:    {p,a3,b3} on C39 by A49,A60,INCSP_1:2;
        p on C1 by A2,A14,Th26;
        then
A63:    {p,b1,a1} on C1 by A48,A59,INCSP_1:2;
A64:    b2 on C2 by A11,A15,Th26;
A65:    a in Line(a,c) by AFF_1:15;
        then
A66:    a1 on A2 by A31,Th26;
A67:    c in Line(a,c) by AFF_1:15;
        then a3 on A2 by A31,Th26;
        then
A68:    {a3,r,a1} on A2 by A37,A66,INCSP_1:2;
A69:    b9 in Line(b9,c9) by AFF_1:15;
        then
A70:    b2 on B1 by A28,Th26;
A71:    c9 in Line(b9,c9) by AFF_1:15;
        then
A72:    b3 on B1 by A28,Th26;
A73:    b3 on B1 by A28,A71,Th26;
A74:    a2 on C2 by A10,A15,Th26;
        then
A75:    {p,a2,b2} on C2 by A34,A64,INCSP_1:2;
A76:    b in Line(b,c) by AFF_1:15;
        then
A77:    a2 on A1 by A29,Th26;
        not p on A1 & not p on B1
        proof
          assume p on A1 or p on B1;
          then a3 on C2 or b3 on C2 by A6,A27,A34,A74,A64,A77,A40,A70,A73,
INCPROJ:def 4;
          hence contradiction by A7,A24,A34,A61,A49,A60,A56,INCPROJ:def 4;
        end;
        then consider t being Element of the Points of XX such that
A78:    t on A1 and
A79:    t on B1 by A34,A61,A74,A64,A49,A60,A77,A40,A70,A73,A56,INCPROJ:def 8;
        a2 on A1 by A29,A76,Th26;
        then
A80:    {a3,a2,t} on A1 by A78,A39,INCSP_1:2;
        b2 on B1 by A28,A69,Th26;
        then
A81:    {t,b2,b3} on B1 by A79,A72,INCSP_1:2;
A82:    a in Line(a,b) by AFF_1:15;
        then
A83:    a1 on A3 by A33,Th26;
A84:    s on A3 by A33,Th30;
A85:    b in Line(a,b) by AFF_1:15;
        then a2 on A3 by A33,Th26;
        then
A86:    {a2,s,a1} on A3 by A84,A83,INCSP_1:2;
        b<>b9 by A2,A3,A5,A6,A8,A9,A10,A14,A15,A17,A19,A22,AFF_4:9;
        then consider O being Element of the Lines of XX such that
A87:    {r,s,t} on O by A1,A5,A6,A7,A22,A23,A27,A24,A47,A63,A75,A62,A80,A68,A86
,A81,A46,A55,A58,INCPROJ:def 13;
A88:    t on O by A87,INCSP_1:2;
A89:    s on O by A87,INCSP_1:2;
A90:    r on O by A87,INCSP_1:2;
A91:    now
          assume
A92:      r<>s;
          ex X st O=[PDir(X),2] & X is being_plane
          proof
            reconsider x=LDir(Line(a,b)),y=LDir(Line(a,c)) as Element of the
            Points of ProjHorizon(AS) by A33,A31,Th14;
A93:        [y,O] in the Inc of IncProjSp_of(AS) by A90,INCSP_1:def 1;
            [x,O] in the Inc of IncProjSp_of(AS) by A89,INCSP_1:def 1;
            then consider
            Z9 being Element of the Lines of ProjHorizon(AS) such
            that
A94:        O=[Z9,2] by A92,A93,Th41;
            consider X such that
A95:        Z9=PDir(X) and
A96:        X is being_plane by Th15;
            take X;
            thus thesis by A94,A95,A96;
          end;
          then not t is Element of AS by A88,Th27;
          then consider Y such that
A97:      t=LDir(Y) and
A98:      Y is being_line by Th20;
          Y '||' Line(b9,c9) by A28,A79,A97,A98,Th28;
          then
A99:      Y // Line(b9,c9) by A28,A98,AFF_4:40;
          Y '||' Line(b,c) by A29,A78,A97,A98,Th28;
          then Y // Line(b,c) by A29,A98,AFF_4:40;
          then Line(b,c) // Line(b9,c9) by A99,AFF_1:44;
          hence thesis by A76,A38,A69,A71,AFF_1:39;
        end;
        now
          assume r=s;
          then
A100:     Line(a,b) // Line(a,c) by A33,A31,Th11;
          then Line(a,b) // Line(a9,c9) by A45,AFF_1:44;
          then Line(a9,b9) // Line(a9, c9) by A54,AFF_1:44;
          then
A101:     c9 in Line(a9,b9) by A52,A43,A41,AFF_1:45;
          c in Line(a,b) by A82,A65,A67,A100,AFF_1:45;
          hence thesis by A85,A50,A54,A101,AFF_1:39;
        end;
        hence thesis by A91;
      end;
      hence
      thesis by A2,A3,A4,A5,A6,A7,A8,A10,A11,A12,A13,A14,A15,A16,A17,A18,A19
,A20,Th50;
    end;
    hence thesis by A10,A11,A12,A13,A15,AFF_1:51;
  end;
  hence thesis by AFF_2:def 4;
end;

theorem
  IncProjSp_of(AS) is Fanoian implies AS is Fanoian
proof
  set XX=IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Fanoian;
  for a,b,c,d being Element of AS st a,b // c,d & a,c // b,d & a,d // b,c
  holds a,b // a,c
  proof
    let a,b,c,d be Element of AS such that
A2: a,b // c,d and
A3: a,c // b,d and
A4: a,d // b,c;
    assume
A5: not a,b // a,c;
    then
A6: a<>d by A2,AFF_1:4;
    then
A7: Line(a,d) is being_line by AFF_1:def 3;
A8: now
      assume b=d;
      then b,a // b,c by A2,AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
    then
A9: Line(b,d) is being_line by AFF_1:def 3;
A10: now
      assume c =d;
      then c,a // c,b by A3,AFF_1:4;
      then LIN c,a,b by AFF_1:def 1;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
    then
A11: Line(c,d) is being_line by AFF_1:def 3;
A12: a<>c by A5,AFF_1:3;
    then
A13: Line(a,c) is being_line by AFF_1:def 3;
A14: a<>b by A5,AFF_1:3;
    then
A15: Line(a,b) is being_line by AFF_1:def 3;
    then reconsider
    a9=LDir(Line(a,b)),b9=LDir(Line(a,c)),c9=LDir(Line(a,d)) as
    Element of the Points of XX by A13,A7,Th20;
A16: b<>c by A5,AFF_1:2;
    then
A17: Line(b,c) is being_line by AFF_1:def 3;
    then reconsider
    L1=[Line(a,b),1],Q1=[Line(c,d),1],R1=[Line(b,d),1],S1=[Line(a,c
),1], A1=[Line(a,d),1],B1=[Line(b,c),1] as Element of the Lines of XX by A15
,A11,A9,A13,A7,Th23;
    reconsider p=a,q=d,r=c,s=b as Element of the Points of XX by Th20;
A18: a9 on L1 by A15,Th30;
    c in Line(b,c) by AFF_1:15;
    then
A19: r on B1 by A17,Th26;
    b in Line(b,c) by AFF_1:15;
    then
A20: s on B1 by A17,Th26;
    Line(a,d) // Line(b,c) by A4,A16,A6,AFF_1:37;
    then Line(a,d) '||' Line(b,c) by A7,A17,AFF_4:40;
    then c9 on B1 by A7,A17,Th28;
    then
A21: {c9,r,s} on B1 by A19,A20,INCSP_1:2;
A22: d in Line(b,d) by AFF_1:15;
    then
A23: q on R1 by A9,Th26;
A24: c in Line(a,c) by AFF_1:15;
    then
A25: r on S1 by A13,Th26;
A26: b9 on S1 by A13,Th30;
A27: a in Line(a,c) by AFF_1:15;
    then p on S1 by A13,Th26;
    then
A28: {b9,p,r} on S1 by A25,A26,INCSP_1:2;
A29: Line(a,c) // Line(b,d) by A3,A12,A8,AFF_1:37;
    then Line(a,c) '||' Line(b,d) by A9,A13,AFF_4:40;
    then
A30: b9 on R1 by A9,A13,Th28;
A31: b in Line(b,d) by AFF_1:15;
    then s on R1 by A9,Th26;
    then
A32: {b9,q,s} on R1 by A23,A30,INCSP_1:2;
A33: now
      assume Line(a,c)=Line(b,d);
      then LIN a,c,b by A31,AFF_1:def 2;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
A34: now
      assume q on S1 or s on S1;
      then d in Line(a,c) or b in Line(a,c) by Th26;
      hence contradiction by A31,A22,A33,A29,AFF_1:45;
    end;
A35: now
      assume p on R1 or r on R1;
      then a in Line(b,d) or c in Line(b,d) by Th26;
      hence contradiction by A27,A24,A33,A29,AFF_1:45;
    end;
A36: a in Line(a,b) by AFF_1:15;
    then consider Y such that
A37: Line(a,b) c= Y and
A38: Line(a,c) c= Y and
A39: Y is being_plane by A27,A15,A13,AFF_4:38;
    reconsider C1=[PDir(Y),2] as Element of the Lines of XX by A39,Th23;
A40: b9 on C1 by A13,A38,A39,Th31;
A41: Line(a,b) // Line(c,d) by A2,A14,A10,AFF_1:37;
    then Line(a,b) '||' Line(c,d) by A15,A11,AFF_4:40;
    then
A42: a9 on Q1 by A15,A11,Th28;
    d in Line(a,d) by AFF_1:15;
    then
A43: q on A1 by A7,Th26;
    a in Line(a,d) by AFF_1:15;
    then
A44: p on A1 by A7,Th26;
    c9 on A1 by A7,Th30;
    then
A45: {c9,p,q} on A1 by A44,A43,INCSP_1:2;
A46: b in Line(a,b) by AFF_1:15;
    then
A47: s on L1 by A15,Th26;
    a9 on C1 by A15,A37,A39,Th31;
    then
A48: {a9,b9} on C1 by A40,INCSP_1:1;
A49: d in Line(c,d) by AFF_1:15;
    then
A50: q on Q1 by A11,Th26;
A51: c in Line(c,d) by AFF_1:15;
    then r on Q1 by A11,Th26;
    then
A52: {a9,q,r} on Q1 by A50,A42,INCSP_1:2;
A53: now
      assume Line(a,b)=Line(c,d);
      then LIN a,b,c by A51,AFF_1:def 2;
      hence contradiction by A5,AFF_1:def 1;
    end;
A54: now
      assume q on L1 or r on L1;
      then d in Line(a,b) or c in Line(a,b) by Th26;
      hence contradiction by A51,A49,A53,A41,AFF_1:45;
    end;
A55: now
      assume p on Q1 or s on Q1;
      then a in Line(c,d) or b in Line(c,d) by Th26;
      hence contradiction by A36,A46,A53,A41,AFF_1:45;
    end;
    Line(b,d)=b*Line(a,c) by A31,A13,A29,AFF_4:def 3;
    then Line(b,d) c= Y by A46,A13,A37,A38,A39,AFF_4:28;
    then
A56: c9 on C1 by A36,A22,A6,A7,A37,A39,Th31,AFF_4:19;
    p on L1 by A36,A15,Th26;
    then {a9,p,s} on L1 by A47,A18,INCSP_1:2;
    hence contradiction by A1,A56,A54,A34,A55,A35,A52,A32,A28,A45,A21,A48,
INCPROJ:def 12;
  end;
  hence thesis by PAPDESAF:def 1;
end;