:: Classical Configurations in Affine Planes
::  by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski

environ

 vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2;
 notations STRUCT_0, ANALOAF, DIRAF, AFF_1;
 constructors AFF_1;
 registrations STRUCT_0;
 theorems AFF_1;

begin

reserve AP for AffinPlane,
  a,a9,b,b9,c,c9,x,y,o,p,q,r,s for Element of AP,
  A,C,C9,D,K,M,N,P,T for Subset of AP;

definition
  let AP;
  attr AP is satisfying_PPAP means

  for M,N,a,b,c,a9,b9,c9 st M is
  being_line & N is being_line & 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;
end;

definition
  let AP be AffinSpace;
  attr AP is Pappian means

  for M,N being Subset of AP, o,a,b,c,a9,b9,c9
being Element of AP 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;
end;

definition
  let AP;
  attr AP is satisfying_PAP_1 means

  for M,N,o,a,b,c,a9,b9,c9 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 & b9 in N & c9 in N & a,b9 // b,
  a9 & b,c9 // c,b9 & a,c9 // c,a9 & b<>c holds a9 in N;
end;

definition
  let AP be AffinSpace;
  attr AP is Desarguesian means

  for A,P,C being Subset of AP, o,a,b,c,
a9,b9,c9 being Element of AP 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;
end;

definition
  let AP;
  attr AP is satisfying_DES_1 means

  for A,P,C,o,a,b,c,a9,b9,c9 st o in
A & o in P & 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 & b,c // b9,c9 & not LIN a,b,c & c <>c9 holds o in
  C;
end;

definition
  let AP;
  attr AP is satisfying_DES_2 means
  for A,P,C,o,a,b,c,a9,b9,c9 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 &
  A is being_line & P is being_line & C is being_line & A<>P & A<>C & a,b //
  a9,b9 & a,c // a9,c9 & b,c // b9,c9 holds c9 in C;
end;

definition
  let AP be AffinSpace;
  attr AP is Moufangian means

  for K being Subset of AP, o,a,b,c,a9,b9,c9 being Element of AP st
  K is being_line & o in K & c in K & c9 in K & not a in K &
  o<>c & a<>b & LIN o,a,a9 & LIN o,b,b9 & a,b // a9,b9 & a,c // a9,c9 & a,
  b // K holds b,c // b9,c9;
end;

definition
  let AP;
  attr AP is satisfying_TDES_1 means

  for K,o,a,b,c,a9,b9,c9 st K is being_line &
  o in K & c in K & c9 in K & not a in K & o<>c & a<>b & LIN o,a,a9
  & a,b // a9,b9 & b,c // b9,c9 & a,c // a9,c9 & a,b // K holds LIN o,b,b9;
end;

definition
  let AP;
  attr AP is satisfying_TDES_2 means

  for K,o,a,b,c,a9,b9,c9 st K is being_line &
  o in K & c in K & c9 in K & not a in K & o<>c & a<>b & LIN o,a,a9
  & LIN o,b,b9 & b,c // b9,c9 & a,c // a9,c9 & a,b // K holds a,b // a9,b9;
end;

definition
  let AP;
  attr AP is satisfying_TDES_3 means

  for K,o,a,b,c,a9,b9,c9 st K is
being_line & o in K & c in K & not a in K & o<>c & a<>b & LIN o,a,a9 & LIN o,b,
  b9 & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & a,b // K holds c9 in K;
end;

definition
  let AP be AffinSpace;
  attr AP is translational means

  for A,P,C being Subset of AP, a,b,c,
  a9,b9,c9 being Element of AP st A // P & A // 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;
end;

definition
  let AP;
  attr AP is satisfying_des_1 means

  for A,P,C,a,b,c,a9,b9,c9 st A // P
& 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 & b
  ,c // b9,c9 & not LIN a,b,c & c <>c9 holds A // C;
end;

definition
  let AP be AffinSpace;
  attr AP is satisfying_pap means

  for M,N being Subset of AP, a,b,c,a9,b9,c9 being Element of AP st
  M is being_line & N is being_line & a in M & b in M & c in M &
  M // N & M<>N & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9
  // c,b9 holds a,c9 // c,a9;
end;

definition
  let AP;
  attr AP is satisfying_pap_1 means

  for M,N,a,b,c,a9,b9,c9 st M is being_line & N is being_line &
  a in M & b in M & c in M & M // N & M<>N & a9 in N & b9 in N &
  a,b9 // b,a9 & b,c9 // c,b9 & a,c9 // c,a9 & a9<>b9 holds c9 in N;
end;

theorem
  AP is Pappian iff AP is satisfying_PAP_1
proof
    hereby assume
A1: AP is Pappian;
    thus AP is satisfying_PAP_1
    proof
      let M,N,o,a,b,c,a9,b9,c9;
      assume 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
      o<>a9 and
A8:  o<>b and
A9:  o<>b9 and
A10:  o<>c and
A11:  o<>c9 and
A12:  a in M and
A13:  b in M and
A14:  c in M and
A15:  b9 in N and
A16:  c9 in N and
A17:  a,b9 // b,a9 and
A18:  b,c9 // c,b9 and
A19:  a,c9 // c,a9 and
A20:  b<>c;
A21:  a<>c9 by A2,A3,A4,A5,A6,A7,A12,A16,AFF_1:18;
A22:  b<>a9
      proof
        assume b=a9;
        then c,b // a,c9 by A19,AFF_1:4;
        then c9 in M by A2,A12,A13,A14,A20,AFF_1:48;
        hence contradiction by A2,A3,A4,A5,A6,A11,A16,AFF_1:18;
      end;
      not b,a9 // N
      proof
        assume
A23:    b,a9 // N;
        b,a9 // a,b9 by A17,AFF_1:4;
        then a,b9 // N by A22,A23,AFF_1:32;
        then b9,a // N by AFF_1:34;
        then a in N by A3,A15,AFF_1:23;
        hence contradiction by A2,A3,A4,A5,A6,A7,A12,AFF_1:18;
      end;
      then consider x such that
A24:  x in N and
A25:  LIN b,a9,x by A3,AFF_1:59;
A26:  b,a9 // b,x by A25,AFF_1:def 1;
A27:  o<>x
      proof
        assume o=x;
        then b,o // a,b9 by A17,A22,A26,AFF_1:5;
        then b9 in M by A2,A5,A8,A12,A13,AFF_1:48;
        hence contradiction by A2,A3,A4,A5,A6,A9,A15,AFF_1:18;
      end;
      a,b9 // b,x by A17,A22,A26,AFF_1:5;
      then a,c9 // c,x by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15
,A16,A18,A24,A27;
      then c,a9 // c,x by A19,A21,AFF_1:5;
      then LIN c,a9,x by AFF_1:def 1;
      then
A28:  LIN a9,x,c by AFF_1:6;
A29:  LIN a9,x,x by AFF_1:7;
      assume
A30:  not a9 in N;
      LIN a9,x,b by A25,AFF_1:6;
      then x in M by A2,A13,A14,A20,A30,A24,A28,A29,AFF_1:8,25;
      hence contradiction by A2,A3,A4,A5,A6,A24,A27,AFF_1:18;
    end;
  end;
    assume
A31: AP is satisfying_PAP_1;
      let M,N,o,a,b,c,a9,b9,c9;
      assume that
A32:   M is being_line and
A33:   N is being_line and
A34:   M<>N & o in M & o in N and
A35:   o<>a and
A36:   o<>a9 and
A37:   o<>b and
A38:   o<>b9 and
A39:  o<>c & o<>c9 and
A40:  a in M and
A41:  b in M and
A42:  c in M and
A43:  a9 in N and
A44:  b9 in N and
A45:  c9 in N and
A46:  a,b9 // b,a9 and
A47:  b,c9 // c,b9;
      set A=Line(a,c9), P=Line(b,a9);
A48:  b<>a9 by A32,A33,A34,A36,A41,A43,AFF_1:18;
      then
A49:  b in P by AFF_1:24;
      assume
A50:  not a,c9 // c,a9;
      then
A51:  a<>c9 by AFF_1:3;
      then
A52:  a in A & c9 in A by AFF_1:24;
A53:  a9 in P by A48,AFF_1:24;
      A is being_line by A51,AFF_1:24;
      then consider K such that
A54:  c in K and
A55:  A // K by AFF_1:49;
A56:  b<>c
      proof
        assume
A57:    b=c;
        then LIN b,c9,b9 by A47,AFF_1:def 1;
        then LIN b9,c9,b by AFF_1:6;
        then b9=c9 or b in N by A33,A44,A45,AFF_1:25;
        hence contradiction by A32,A33,A34,A37,A41,A46,A50,A57,AFF_1:18;
      end;
A58:  b9<>c9
      proof
        assume b9=c9;
        then b9,b // b9,c by A47,AFF_1:4;
        then LIN b9,b,c by AFF_1:def 1;
        then LIN b,c,b9 by AFF_1:6;
        then b9 in M by A32,A41,A42,A56,AFF_1:25;
        hence contradiction by A32,A33,A34,A38,A44,AFF_1:18;
      end;
A59:  not P // K
      proof
        assume P // K;
        then P // A by A55,AFF_1:44;
        then b,a9 // a,c9 by A52,A49,A53,AFF_1:39;
        then a,b9 // a,c9 by A46,A48,AFF_1:5;
        then LIN a,b9,c9 by AFF_1:def 1;
        then LIN b9,c9,a by AFF_1:6;
        then a in N by A33,A44,A45,A58,AFF_1:25;
        hence contradiction by A32,A33,A34,A35,A40,AFF_1:18;
      end;
A60:  P is being_line by A48,AFF_1:24;
      K is being_line by A55,AFF_1:36;
      then consider x such that
A61:  x in P and
A62:  x in K by A60,A59,AFF_1:58;
A63:  a,c9 // c,x by A52,A54,A55,A62,AFF_1:39;
      LIN b,x,a9 by A60,A49,A53,A61,AFF_1:21;
      then b,x // b,a9 by AFF_1:def 1;
      then a,b9 // b,x by A46,A48,AFF_1:5;
      then x in N by A31,A32,A33,A34,A35,A37,A38,A39,A40,A41,A42,A44,A45,A47
,A56,A63;
      then N=P by A33,A43,A50,A60,A53,A61,A63,AFF_1:18;
      hence contradiction by A32,A33,A34,A37,A41,A49,AFF_1:18;
end;

theorem
  AP is Desarguesian iff AP is satisfying_DES_1
proof
    hereby assume
A1: AP is Desarguesian;
    thus AP is satisfying_DES_1
    proof
      let A,P,C,o,a,b,c,a9,b9,c9;
      assume that
A2:  o in A and
A3:  o in P and
A4:  o<>a and
A5:  o<>b and
      o<>c and
A6:  a in A and
A7:  a9 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:  b,c // b9,c9 and
A20:  not LIN a,b,c and
A21:  c <>c9;
      set K=Line(o,c9);
      assume
A22:  not o in C;
      then
A23:  K is being_line by A11,AFF_1:24;
A24:  a9<>c9
      proof
        assume
A25:    a9=c9;
        then b,c // a9,b9 by A19,AFF_1:4;
        then a,b // b,c or a9=b9 by A17,AFF_1:5;
        then b,a // b,c or a9=b9 by AFF_1:4;
        then LIN b,a,c or a9=b9 by AFF_1:def 1;
        hence contradiction by A2,A3,A7,A9,A11,A12,A13,A15,A20,A22,A25,AFF_1:6
,18;
      end;
A26:  A<>K
      proof
        assume A=K;
        then
A27:    c9 in A by A2,AFF_1:24;
        a9,c9 // a,c by A18,AFF_1:4;
        then c in A by A6,A7,A12,A24,A27,AFF_1:48;
        hence contradiction by A10,A11,A12,A14,A16,A21,A27,AFF_1:18;
      end;
A28:  a<>c by A20,AFF_1:7;
A29:  o in K by A11,A22,AFF_1:24;
A30:  c9 in K by A11,A22,AFF_1:24;
      not a,c // K
      proof
        assume a,c // K;
        then a9,c9 // K by A18,A28,AFF_1:32;
        then c9,a9 // K by AFF_1:34;
        then a9 in K by A23,A30,AFF_1:23;
        then
A31:    a9 in P by A2,A3,A7,A12,A23,A29,A26,AFF_1:18;
        a9,b9 // b,a by A17,AFF_1:4;
        then a9=b9 or a in P by A8,A9,A13,A31,AFF_1:48;
        then a,c // b,c by A2,A3,A4,A6,A12,A13,A15,A18,A19,A24,AFF_1:5,18;
        then c,a // c,b by AFF_1:4;
        then LIN c,a,b by AFF_1:def 1;
        hence contradiction by A20,AFF_1:6;
      end;
      then consider x such that
A32:  x in K and
A33:  LIN a,c,x by A23,AFF_1:59;
A34:  o<>x
      proof
        assume o=x;
        then LIN a,o,c by A33,AFF_1:6;
        then
A35:    c in A by A2,A4,A6,A12,AFF_1:25;
        then c9 in A by A6,A7,A12,A18,A28,AFF_1:48;
        hence contradiction by A10,A11,A12,A14,A16,A21,A35,AFF_1:18;
      end;
A36:  b9<>c9
      proof
        assume b9=c9;
        then a9=b9 or a,c // a,b by A17,A18,AFF_1:5;
        then a9=b9 or LIN a,c,b by AFF_1:def 1;
        then b,c // a,c by A18,A19,A20,A24,AFF_1:5,6;
        then c,b // c,a by AFF_1:4;
        then LIN c,b,a by AFF_1:def 1;
        hence contradiction by A20,AFF_1:6;
      end;
A37:  a,c // a,x by A33,AFF_1:def 1;
      then a,x // a9,c9 by A18,A28,AFF_1:5;
      then
b,x // b9,c9 by A1,A2,A3,A4,A5,A6,A7,A8,A9,A12,A13,A15,A17,A23,A29,A30,A26,A32
,A34;
      then
A38:  b,x // b,c by A19,A36,AFF_1:5;
A39:  not LIN a,b,x
      proof
        assume LIN a,b,x;
        then a,b // a,x by AFF_1:def 1;
        then a,b // a,c or a=x by A37,AFF_1:5;
        hence contradiction by A2,A4,A6,A12,A20,A23,A29,A26,A32,AFF_1:18,def 1;
      end;
      LIN a,x,c by A33,AFF_1:6;
      then c in K by A32,A39,A38,AFF_1:14;
      hence contradiction by A10,A11,A14,A21,A22,A23,A29,A30,AFF_1:18;
    end;
  end;
    assume
A40: AP is satisfying_DES_1;
      let A,P,C,o,a,b,c,a9,b9,c9;
      assume that
A41:   o in A and
A42:   o in P and
A43:   o in C and
A44:   o<>a and
A45:   o<>b and
A46:   o<>c and
A47:   a in A and
A48:  a9 in A and
A49:  b in P and
A50:  b9 in P and
A51:  c in C and
A52:  c9 in C and
A53:  A is being_line and
A54:  P is being_line and
A55:  C is being_line and
A56:  A<>P and
A57:  A<>C and
A58:  a,b // a9,b9 and
A59:  a,c // a9,c9;
      assume
A60:  not b,c // b9,c9;
A61:  a9<>b9
      proof
A62:    a9,c9 // c,a by A59,AFF_1:4;
        assume
A63:    a9=b9;
        then a9 in C by A41,A42,A43,A48,A50,A53,A54,A56,AFF_1:18;
        then a in C or a9=c9 by A51,A52,A55,A62,AFF_1:48;
        hence contradiction by A41,A43,A44,A47,A53,A55,A57,A60,A63,AFF_1:3,18;
      end;
A64:  a9<>c9
      proof
        assume a9=c9;
        then
A65:    a9 in P by A41,A42,A43,A48,A52,A53,A55,A57,AFF_1:18;
        a9,b9 // b,a by A58,AFF_1:4;
        then a in P by A49,A50,A54,A61,A65,AFF_1:48;
        hence contradiction by A41,A42,A44,A47,A53,A54,A56,AFF_1:18;
      end;
A66:  o<>c9
      proof
        assume
A67:    o=c9;
        a9,c9 // a,c by A59,AFF_1:4;
        then c in A by A41,A47,A48,A53,A64,A67,AFF_1:48;
        hence contradiction by A41,A43,A46,A51,A53,A55,A57,AFF_1:18;
      end;
      set M=Line(a,c), N=Line(b9,c9);
A68:  a<>c by A41,A43,A44,A47,A51,A53,A55,A57,AFF_1:18;
      then
A69:  c in M by AFF_1:24;
A70:  a<>b by A41,A42,A44,A47,A49,A53,A54,A56,AFF_1:18;
A71:  not LIN a9,b9,c9
      proof
        assume
A72:    LIN a9,b9,c9;
        then a9,b9 // a9,c9 by AFF_1:def 1;
        then a9,b9 // a,c by A59,A64,AFF_1:5;
        then a,b // a,c by A58,A61,AFF_1:5;
        then LIN a,b,c by AFF_1:def 1;
        then LIN b,c,a by AFF_1:6;
        then b,c // b,a by AFF_1:def 1;
        then b,c // a,b by AFF_1:4;
        then
A73:    b,c // a9,b9 by A58,A70,AFF_1:5;
        LIN b9,c9,a9 by A72,AFF_1:6;
        then b9,c9 // b9,a9 by AFF_1:def 1;
        then b9,c9 // a9,b9 by AFF_1:4;
        hence contradiction by A60,A61,A73,AFF_1:5;
      end;
A74:  b9<>c9 by A60,AFF_1:3;
      then
A75:  b9 in N & c9 in N by AFF_1:24;
      N is being_line by A74,AFF_1:24;
      then consider D such that
A76:  b in D and
A77:  N // D by AFF_1:49;
A78:  a in M by A68,AFF_1:24;
A79:  not M // D
      proof
        assume M // D;
        then M // N by A77,AFF_1:44;
        then a,c // b9,c9 by A78,A69,A75,AFF_1:39;
        then a9,c9 // b9,c9 by A59,A68,AFF_1:5;
        then c9,a9 // c9,b9 by AFF_1:4;
        then LIN c9,a9,b9 by AFF_1:def 1;
        hence contradiction by A71,AFF_1:6;
      end;
A80:  M is being_line by A68,AFF_1:24;
      D is being_line by A77,AFF_1:36;
      then consider x such that
A81:  x in M and
A82:  x in D by A80,A79,AFF_1:58;
      LIN a,c,x by A80,A78,A69,A81,AFF_1:21;
      then a,c // a,x by AFF_1:def 1;
      then
A83:  a,x // a9,c9 by A59,A68,AFF_1:5;
      set T=Line(c9,x);
A84:  a<>a9
      proof
        assume
A85:    a=a9;
        then LIN a,c,c9 by A59,AFF_1:def 1;
        then LIN c,c9,a by AFF_1:6;
        then
A86:    c =c9 or a in C by A51,A52,A55,AFF_1:25;
        LIN a,b,b9 by A58,A85,AFF_1:def 1;
        then LIN b,b9,a by AFF_1:6;
        then b=b9 or a in P by A49,A50,A54,AFF_1:25;
        hence contradiction by A41,A42,A43,A44,A47,A53,A54,A55,A56,A57,A60,A86,
AFF_1:2,18;
      end;
A87:  x<>c9
      proof
        assume x=c9;
        then c9,a // c9,a9 by A83,AFF_1:4;
        then LIN c9,a,a9 by AFF_1:def 1;
        then LIN a,a9,c9 by AFF_1:6;
        then c9 in A by A47,A48,A53,A84,AFF_1:25;
        hence contradiction by A41,A43,A52,A53,A55,A57,A66,AFF_1:18;
      end;
      then
A88:  T is being_line & c9 in T by AFF_1:24;
A89:  b,x // b9,c9 by A75,A76,A77,A82,AFF_1:39;
A90:  x in T by A87,AFF_1:24;
A91:  a<>x
      proof
        assume a=x;
        then a,b // b9,c9 by A75,A76,A77,A82,AFF_1:39;
        then a9,b9 // b9,c9 by A58,A70,AFF_1:5;
        then b9,a9 // b9,c9 by AFF_1:4;
        then LIN b9,a9,c9 by AFF_1:def 1;
        hence contradiction by A71,AFF_1:6;
      end;
      not LIN a,b,x
      proof
        assume LIN a,b,x;
        then a,b // a,x by AFF_1:def 1;
        then a,b // a9,c9 by A83,A91,AFF_1:5;
        then a9,b9 // a9,c9 by A58,A70,AFF_1:5;
        hence contradiction by A71,AFF_1:def 1;
      end;
      then o in T by A40,A41,A42,A44,A45,A47,A48,A49,A50,A53,A54,A56,A58,A83
,A89,A87,A88,A90;
      then x in C by A43,A52,A55,A66,A88,A90,AFF_1:18;
      then C=M by A51,A55,A60,A80,A69,A81,A89,AFF_1:18;
      hence contradiction by A41,A43,A44,A47,A53,A55,A57,A78,AFF_1:18;
end;

theorem Th3:
  AP is Moufangian implies AP is satisfying_TDES_1
proof
  assume
A1: AP is Moufangian;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: c9 in K and
A6: not a in K and
A7: o<>c and
A8: a<>b and
A9: LIN o,a,a9 and
A10: a,b // a9,b9 and
A11: b,c // b9,c9 and
A12: a,c // a9,c9 and
A13: a,b // K;
    consider P such that
A14: a9 in P and
A15: K // P by A2,AFF_1:49;
A16: P is being_line by A15,AFF_1:36;
    set A=Line(o,b), C=Line(o,a);
A17: o in A & b in A by AFF_1:15;
    assume
A18: not LIN o,b,b9;
    then o<>b by AFF_1:7;
    then
A19: A is being_line by AFF_1:def 3;
A20: not b in K by A6,A13,AFF_1:35;
    not A // P
    proof
      assume A // P;
      then A // K by A15,AFF_1:44;
      hence contradiction by A3,A20,A17,AFF_1:45;
    end;
    then consider x such that
A21: x in A and
A22: x in P by A19,A16,AFF_1:58;
A23: o in C & a in C by AFF_1:15;
A24: LIN o,b,x by A19,A17,A21,AFF_1:21;
    a,b // P by A13,A15,AFF_1:43;
    then a9,b9 // P by A8,A10,AFF_1:32;
    then
A25: b9 in P by A14,A16,AFF_1:23;
    then
A26: LIN b9,x,b9 by A16,A22,AFF_1:21;
A27: C is being_line by A3,A6,AFF_1:def 3;
    then
A28: a9 in C by A3,A6,A9,A23,AFF_1:25;
A29: b9<>c9
    proof
A30:  a9,c9 // c,a by A12,AFF_1:4;
      assume
A31:  b9=c9;
      then P=K by A5,A15,A25,AFF_1:45;
      then a9=o by A2,A3,A6,A27,A23,A28,A14,AFF_1:18;
      then b9=o by A2,A3,A4,A5,A6,A31,A30,AFF_1:48;
      hence contradiction by A18,AFF_1:7;
    end;
A32: b<>c by A4,A6,A13,AFF_1:35;
    a9,x // K by A14,A15,A22,AFF_1:40;
    then a,b // a9,x by A2,A13,AFF_1:31;
    then b,c // x,c9 by A1,A2,A3,A4,A5,A6,A7,A8,A9,A12,A13,A24;
    then b9,c9 // x,c9 by A11,A32,AFF_1:5;
    then c9,b9 // c9,x by AFF_1:4;
    then LIN c9,b9,x by AFF_1:def 1;
    then
A33: LIN b9,x,c9 by AFF_1:6;
A34: a9<>b9
    proof
      assume
A35:  a9=b9;
A36:  now
        assume a9=c9;
        then b9=o by A2,A3,A5,A6,A27,A23,A28,A35,AFF_1:18;
        hence contradiction by A18,AFF_1:7;
      end;
      a,c // b,c or a9=c9 by A11,A12,A35,AFF_1:5;
      then c,a // c,b by A36,AFF_1:4;
      then LIN c,a,b by AFF_1:def 1;
      then LIN a,c,b by AFF_1:6;
      then a,c // a,b by AFF_1:def 1;
      then a,b // a,c by AFF_1:4;
      then a,c // K by A8,A13,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A6,AFF_1:23;
    end;
    LIN b9,x,a9 by A14,A16,A22,A25,AFF_1:21;
    then LIN b9,c9,a9 by A18,A24,A33,A26,AFF_1:8;
    then b9,c9 // b9,a9 by AFF_1:def 1;
    then b9,c9 // a9,b9 by AFF_1:4;
    then b,c // a9,b9 by A11,A29,AFF_1:5;
    then a,b // b,c by A10,A34,AFF_1:5;
    then b,c // K by A8,A13,AFF_1:32;
    then c,b // K by AFF_1:34;
    hence contradiction by A2,A4,A20,AFF_1:23;
end;

theorem
  AP is satisfying_TDES_1 implies AP is satisfying_TDES_2
proof
  assume
A1: AP is satisfying_TDES_1;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: c9 in K and
A6: not a in K and
A7: o<>c and
A8: a<>b and
A9: LIN o,a,a9 and
A10: LIN o,b,b9 and
A11: b,c // b9,c9 and
A12: a,c // a9,c9 and
A13: a,b // K;
    set A=Line(o,a), P=Line(o,b);
A14: A is being_line & a in A by A3,A6,AFF_1:24;
A15: o in A by A3,A6,AFF_1:24;
    then
A16: a9 in A by A3,A6,A9,A14,AFF_1:25;
A17: o<>b by A3,A6,A13,AFF_1:35;
    then
A18: P is being_line by AFF_1:24;
    consider N such that
A19: a9 in N and
A20: K // N by A2,AFF_1:49;
A21: N is being_line by A20,AFF_1:36;
    set T=Line(b9,c9);
A22: not b in K by A6,A13,AFF_1:35;
A23: b in P by A17,AFF_1:24;
A24: o in P by A17,AFF_1:24;
    then
A25: b9 in P by A10,A17,A18,A23,AFF_1:25;
    assume
A26: not a,b // a9,b9;
    then
A27: a9<>b9 by AFF_1:3;
A28: not b9 in K
    proof
A29:  a9,c9 // a,c by A12,AFF_1:4;
A30:  b9,c9 // c,b by A11,AFF_1:4;
      assume
A31:  b9 in K;
      then b9=o by A2,A3,A22,A18,A24,A23,A25,AFF_1:18;
      then c9 in A by A2,A3,A4,A5,A22,A15,A30,AFF_1:48;
      then a9=c9 or c in A by A14,A16,A29,AFF_1:48;
      hence contradiction by A2,A3,A4,A5,A6,A7,A27,A22,A15,A14,A31,A30,AFF_1:18
,48;
    end;
    then
A32: T is being_line by A5,AFF_1:24;
A33: b9 in T by A5,A28,AFF_1:24;
A34: c9 in T by A5,A28,AFF_1:24;
    not N // T
    proof
      assume N // T;
      then K // T by A20,AFF_1:44;
      hence contradiction by A5,A28,A33,A34,AFF_1:45;
    end;
    then consider x such that
A35: x in N and
A36: x in T by A32,A21,AFF_1:58;
    a9,x // K by A19,A20,A35,AFF_1:40;
    then
A37: a,b // a9,x by A2,A13,AFF_1:31;
    LIN c9,b9,x by A32,A33,A34,A36,AFF_1:21;
    then c9,b9 // c9,x by AFF_1:def 1;
    then b9,c9 // x,c9 by AFF_1:4;
    then b,c // x,c9 by A5,A11,A28,AFF_1:5;
    then LIN o,b,x by A1,A2,A3,A4,A5,A6,A7,A8,A9,A12,A13,A37;
    then x in P by A17,A18,A24,A23,AFF_1:25;
    then P=T by A26,A18,A25,A32,A33,A36,A37,AFF_1:18;
    then LIN c9,b9,b by A18,A23,A33,A34,AFF_1:21;
    then c9,b9 // c9,b by AFF_1:def 1;
    then b9,c9 // b,c9 by AFF_1:4;
    then b,c // b,c9 by A5,A11,A28,AFF_1:5;
    then LIN b,c,c9 by AFF_1:def 1;
    then
A38: LIN c,c9,b by AFF_1:6;
    then a,c // a9,c by A2,A4,A5,A12,A22,AFF_1:25;
    then c,a // c,a9 by AFF_1:4;
    then LIN c,a,a9 by AFF_1:def 1;
    then LIN a,a9,c by AFF_1:6;
    then
A39: a=a9 or c in A by A14,A16,AFF_1:25;
    b,c // b9,c by A2,A4,A5,A11,A22,A38,AFF_1:25;
    then c,b // c,b9 by AFF_1:4;
    then LIN c,b,b9 by AFF_1:def 1;
    then LIN b,b9,c by AFF_1:6;
    then b=b9 or c in P by A18,A23,A25,AFF_1:25;
    hence contradiction by A2,A3,A4,A6,A7,A26,A22,A18,A15,A14,A24,A23,A39,
AFF_1:2,18;
end;

theorem
  AP is satisfying_TDES_2 implies AP is satisfying_TDES_3
proof
  assume
A1: AP is satisfying_TDES_2;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: not a in K and
A6: o<>c and
A7: a<>b and
A8: LIN o,a,a9 and
A9: LIN o,b,b9 and
A10: a,b // a9,b9 and
A11: a,c // a9,c9 and
A12: b,c // b9,c9 and
A13: a,b // K;
    set A=Line(o,a), P=Line(o,b), N=Line(b,c);
A14: o in A by A3,A5,AFF_1:24;
A15: not LIN a,b,c
    proof
      assume LIN a,b,c;
      then a,b // a,c by AFF_1:def 1;
      then a,c // K by A7,A13,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A5,AFF_1:23;
    end;
A16: o<>b by A3,A5,A13,AFF_1:35;
    then
A17: b in P by AFF_1:24;
A18: a9,b9 // b,a by A10,AFF_1:4;
A19: b<>c by A4,A5,A13,AFF_1:35;
    then
A20: b in N & c in N by AFF_1:24;
A21: a in A by A3,A5,AFF_1:24;
A22: A is being_line by A3,A5,AFF_1:24;
A23: A<>P
    proof
      assume A=P;
      then a,b // A by A22,A21,A17,AFF_1:40,41;
      hence contradiction by A3,A5,A7,A13,A14,A21,AFF_1:45,53;
    end;
    assume
A24: not c9 in K;
A25: P is being_line by A16,AFF_1:24;
A26: o in P by A16,AFF_1:24;
    then
A27: b9 in P by A9,A16,A25,A17,AFF_1:25;
A28: a9 in A by A3,A5,A8,A22,A14,A21,AFF_1:25;
A29: a9<>b9
    proof
      assume
A30:  a9=b9;
      then a,c // b,c or a9=c9 by A11,A12,AFF_1:5;
      then c,a // c,b or a9=c9 by AFF_1:4;
      then LIN c,a,b or a9=c9 by AFF_1:def 1;
      hence contradiction by A3,A24,A15,A22,A25,A14,A26,A28,A27,A23,A30,AFF_1:6
,18;
    end;
A31: a9<>c9
    proof
      assume a9=c9;
      then b,c // a9,b9 by A12,AFF_1:4;
      then a,b // b,c by A10,A29,AFF_1:5;
      then b,a // b,c by AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      hence contradiction by A15,AFF_1:6;
    end;
    not a9,c9 // K
    proof
      assume
A32:  a9,c9 // K;
      a9,c9 // a,c by A11,AFF_1:4;
      then a,c // K by A31,A32,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A5,AFF_1:23;
    end;
    then consider x such that
A33: x in K and
A34: LIN a9,c9,x by A2,AFF_1:59;
    a9,c9 // a9,x by A34,AFF_1:def 1;
    then
A35: a,c // a9,x by A11,A31,AFF_1:5;
    N is being_line by A19,AFF_1:24;
    then consider T such that
A36: x in T and
A37: N // T by AFF_1:49;
A38: not b in K by A5,A13,AFF_1:35;
A39: not T // P
    proof
      assume T // P;
      then N // P by A37,AFF_1:44;
      then c in P by A17,A20,AFF_1:45;
      hence contradiction by A2,A3,A4,A6,A38,A25,A26,A17,AFF_1:18;
    end;
    T is being_line by A37,AFF_1:36;
    then consider y such that
A40: y in T and
A41: y in P by A25,A39,AFF_1:58;
A42: b,c // y,x by A20,A36,A37,A40,AFF_1:39;
A43: now
      assume y=b9;
      then b9,c9 // b9,x by A12,A19,A42,AFF_1:5;
      then LIN b9,c9,x by AFF_1:def 1;
      then
A44:  LIN c9,x,b9 by AFF_1:6;
      LIN c9,x,a9 & LIN c9,x,c9 by A34,AFF_1:6,7;
      then LIN a9,b9,c9 by A24,A33,A44,AFF_1:8;
      then a9,b9 // a9,c9 by AFF_1:def 1;
      then a9,b9 // a,c by A11,A31,AFF_1:5;
      then a,b // a,c by A10,A29,AFF_1:5;
      hence contradiction by A15,AFF_1:def 1;
    end;
    LIN o,b,y by A25,A26,A17,A41,AFF_1:21;
    then a,b // a9,y by A1,A2,A3,A4,A5,A6,A7,A8,A13,A33,A42,A35;
    then a9,b9 // a9,y by A7,A10,AFF_1:5;
    then LIN a9,b9,y by AFF_1:def 1;
    then LIN b9,y,a9 by AFF_1:6;
    then a9 in P by A25,A27,A41,A43,AFF_1:25;
    then a in P by A25,A17,A27,A29,A18,AFF_1:48;
    hence contradiction by A3,A5,A25,A26,A23,AFF_1:24;
end;

theorem
  AP is satisfying_TDES_3 implies AP is Moufangian
proof
  assume
A1: AP is satisfying_TDES_3;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: c9 in K and
A6: not a in K and
A7: o<>c and
A8: a<>b and
A9: LIN o,a,a9 and
A10: LIN o,b,b9 and
A11: a,b // a9,b9 and
A12: a,c // a9,c9 and
A13: a,b // K;
    set A=Line(o,a), P=Line(o,b), M=Line(b,c), T=Line(a9,c9);
A14: o in A by A3,A6,AFF_1:24;
    assume
A15: not b,c // b9,c9;
    then
A16: b<>c by AFF_1:3;
    then
A17: b in M by AFF_1:24;
A18: a9,b9 // b,a by A11,AFF_1:4;
A19: c in M by A16,AFF_1:24;
A20: a in A by A3,A6,AFF_1:24;
A21: A is being_line by A3,A6,AFF_1:24;
    then
A22: a9 in A by A3,A6,A9,A14,A20,AFF_1:25;
A23: o<>b by A3,A6,A13,AFF_1:35;
    then
A24: P is being_line by AFF_1:24;
A25: b in P by A23,AFF_1:24;
A26: A<>P
    proof
      assume A=P;
      then a,b // A by A21,A20,A25,AFF_1:40,41;
      hence contradiction by A3,A6,A8,A13,A14,A20,AFF_1:45,53;
    end;
A27: o in P by A23,AFF_1:24;
    then
A28: b9 in P by A10,A23,A24,A25,AFF_1:25;
A29: a9<>b9
    proof
A30:  a9,c9 // c,a by A12,AFF_1:4;
      assume
A31:  a9=b9;
      then a9 in K by A3,A21,A24,A14,A27,A22,A28,A26,AFF_1:18;
      then a9=c9 by A2,A4,A5,A6,A30,AFF_1:48;
      hence contradiction by A15,A31,AFF_1:3;
    end;
A32: a9<>c9
    proof
      assume a9=c9;
      then
A33:  a9 in P by A2,A3,A5,A6,A21,A14,A20,A27,A22,AFF_1:18;
      a9,b9 // b,a by A11,AFF_1:4;
      then a in P by A24,A25,A28,A29,A33,AFF_1:48;
      hence contradiction by A3,A6,A24,A27,A26,AFF_1:24;
    end;
    then
A34: T is being_line & c9 in T by AFF_1:24;
A35: M is being_line by A16,AFF_1:24;
    then consider N such that
A36: b9 in N and
A37: M // N by AFF_1:49;
A38: N is being_line by A37,AFF_1:36;
A39: not LIN a,b,c
    proof
      assume LIN a,b,c;
      then a,b // a,c by AFF_1:def 1;
      then a,c // K by A8,A13,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A6,AFF_1:23;
    end;
    not a9,c9 // N
    proof
      assume
A40:  a9,c9 // N;
      a9,c9 // a,c by A12,AFF_1:4;
      then a,c // N by A32,A40,AFF_1:32;
      then a,c // M by A37,AFF_1:43;
      then c,a // M by AFF_1:34;
      then a in M by A35,A19,AFF_1:23;
      hence contradiction by A39,A35,A17,A19,AFF_1:21;
    end;
    then consider x such that
A41: x in N and
A42: LIN a9,c9,x by A38,AFF_1:59;
A43: b,c // b9,x by A17,A19,A36,A37,A41,AFF_1:39;
    a9,c9 // a9,x by A42,AFF_1:def 1;
    then a,c // a9,x by A12,A32,AFF_1:5;
    then
A44: x in K by A1,A2,A3,A4,A6,A7,A8,A9,A10,A11,A13,A43;
A45: a9 in T by A32,AFF_1:24;
    then x in T by A32,A34,A42,AFF_1:25;
    then K=T by A2,A5,A15,A34,A43,A44,AFF_1:18;
    then a9 in P by A2,A3,A6,A21,A14,A20,A27,A22,A45,AFF_1:18;
    then a in P by A24,A25,A28,A29,A18,AFF_1:48;
    hence contradiction by A3,A6,A24,A27,A26,AFF_1:24;
end;

theorem Th7:
  AP is translational iff AP is satisfying_des_1
proof
    hereby assume
A1: AP is translational;
     thus AP is satisfying_des_1
     proof
      let A,P,C,a,b,c,a9,b9,c9;
      assume that
A2:  A // P and
A3:  a in A and
A4:  a9 in A and
A5:  b in P and
A6:  b9 in P and
A7:  c in C & c9 in C and
A8:  A is being_line and
A9:  P is being_line and
A10:  C is being_line and
A11:  A<>P and
A12:  A<>C and
A13:  a,b // a9,b9 and
A14:  a,c // a9,c9 and
A15:  b,c // b9,c9 and
A16:  not LIN a,b,c and
A17:  c <>c9;
      assume
A18:  not A // C;
      consider K such that
A19:  c9 in K and
A20:  A // K by A8,AFF_1:49;
A21:  a<>c by A16,AFF_1:7;
A22:  not a,c // K
      proof
        assume a,c // K;
        then a,c // A by A20,AFF_1:43;
        then
A23:    c in A by A3,A8,AFF_1:23;
        a9,c9 // a,c by A14,AFF_1:4;
        then a9,c9 // A by A3,A8,A21,A23,AFF_1:27;
        then c9 in A by A4,A8,AFF_1:23;
        hence contradiction by A7,A8,A10,A12,A17,A23,AFF_1:18;
      end;
A24:  A<>K
      proof
        assume
A25:    A=K;
        a9,c9 // a,c by A14,AFF_1:4;
        then a9=c9 by A4,A19,A20,A22,A25,AFF_1:32,40;
        then a9,b9 // b,c by A15,AFF_1:4;
        then a9=b9 or a,b // b,c by A13,AFF_1:5;
        then b9 in A or b,a // b,c by A4,AFF_1:4;
        then LIN b,a,c by A2,A6,A11,AFF_1:45,def 1;
        hence contradiction by A16,AFF_1:6;
      end;
A26:  now
        assume b9=c9;
        then a,b // a,c or a9=b9 by A13,A14,AFF_1:5;
        hence contradiction by A2,A4,A6,A11,A16,AFF_1:45,def 1;
      end;
A27:  K is being_line by A20,AFF_1:36;
      then consider x such that
A28:  x in K and
A29:  LIN a,c,x by A22,AFF_1:59;
      a,c // a,x by A29,AFF_1:def 1;
      then a,x // a9,c9 by A14,A21,AFF_1:5;
      then b,x // b9,c9 by A1,A2,A3,A4,A5,A6,A8,A9,A11,A13,A19,A20,A27,A28,A24;
      then b,x // b,c or b9=c9 by A15,AFF_1:5;
      then LIN b,x,c by A26,AFF_1:def 1;
      then
A30:  LIN x,c,b by AFF_1:6;
A31:  LIN x,c, c by AFF_1:7;
      LIN x,c,a by A29,AFF_1:6;
      then c in K by A16,A28,A30,A31,AFF_1:8;
      hence contradiction by A7,A10,A17,A18,A19,A20,A27,AFF_1:18;
    end;
  end;
    assume
A32: AP is satisfying_des_1;
      let A,P,C,a,b,c,a9,b9,c9;
      assume that
A33:   A // P and
A34:   A // C and
A35:   a in A and
A36:   a9 in A and
A37:   b in P and
A38:   b9 in P and
A39:   c in C and
A40:  c9 in C and
A41:  A is being_line and
A42:  P is being_line and
A43:  C is being_line and
A44:  A<>P and
A45:  A<>C and
A46:  a,b // a9,b9 and
A47:  a,c // a9,c9;
A48:  a<>b by A33,A35,A37,A44,AFF_1:45;
A49:  a9<>b9 by A33,A36,A38,A44,AFF_1:45;
      set K=Line(a,c), N=Line(b9,c9);
A50:  a<>c by A34,A35,A39,A45,AFF_1:45;
      then
A51:  a in K by AFF_1:24;
      assume
A52:  not b,c // b9,c9;
      then
A53:  b9<>c9 by AFF_1:3;
      then
A54:  b9 in N by AFF_1:24;
A55:  b<>c by A52,AFF_1:3;
A56:  not LIN a,b,c
      proof
        assume
A57:    LIN a,b,c;
        then LIN b,c,a by AFF_1:6;
        then b,c // b,a by AFF_1:def 1;
        then b,c // a,b by AFF_1:4;
        then
A58:    b,c // a9,b9 by A46,A48,AFF_1:5;
        LIN c,b,a by A57,AFF_1:6;
        then c,b // c,a by AFF_1:def 1;
        then b,c // a,c by AFF_1:4;
        then b,c // a9,c9 by A47,A50,AFF_1:5;
        then a9,c9 // a9,b9 by A55,A58,AFF_1:5;
        then LIN a9,c9,b9 by AFF_1:def 1;
        then LIN b9,c9,a9 by AFF_1:6;
        then b9,c9 // b9,a9 by AFF_1:def 1;
        then b9,c9 // a9,b9 by AFF_1:4;
        hence contradiction by A52,A49,A58,AFF_1:5;
      end;
A59:  c in K by A50,AFF_1:24;
A60:  N is being_line by A53,AFF_1:24;
      then consider M such that
A61:  b in M and
A62:  N // M by AFF_1:49;
A63:  c9 in N by A53,AFF_1:24;
A64:  a9<>c9 by A34,A36,A40,A45,AFF_1:45;
A65:  not LIN a9,b9,c9
      proof
        assume LIN a9,b9,c9;
        then a9,b9 // a9,c9 by AFF_1:def 1;
        then a,b // a9,c9 by A46,A49,AFF_1:5;
        then a,b // a,c by A47,A64,AFF_1:5;
        hence contradiction by A56,AFF_1:def 1;
      end;
A66:  not K // M
      proof
        assume K // M;
        then K // N by A62,AFF_1:44;
        then a,c // b9,c9 by A51,A59,A54,A63,AFF_1:39;
        then a9,c9 // b9,c9 by A47,A50,AFF_1:5;
        then c9,a9 // c9,b9 by AFF_1:4;
        then LIN c9,a9,b9 by AFF_1:def 1;
        hence contradiction by A65,AFF_1:6;
      end;
A67:  K is being_line by A50,AFF_1:24;
A68:  M is being_line by A62,AFF_1:36;
      then consider x such that
A69:  x in K and
A70:  x in M by A67,A66,AFF_1:58;
A71:  b,x // b9,c9 by A54,A63,A61,A62,A70,AFF_1:39;
      set D=Line(x,c9);
A72:  A<>D
      proof
        assume A=D;
        then c9 in A by AFF_1:15;
        hence contradiction by A34,A40,A45,AFF_1:45;
      end;
A73:  x in D by AFF_1:15;
      LIN a,c,x by A67,A51,A59,A69,AFF_1:21;
      then a,c // a,x by AFF_1:def 1;
      then
A74:  a,x // a9,c9 by A47,A50,AFF_1:5;
A75:  c9 in D by AFF_1:15;
A76:  not LIN a,b,x
      proof
A77:    a<>x
        proof
          assume a=x;
          then a,b // b9,c9 by A54,A63,A61,A62,A70,AFF_1:39;
          then a9,b9 // b9,c9 by A46,A48,AFF_1:5;
          then b9,a9 // b9,c9 by AFF_1:4;
          then LIN b9,a9,c9 by AFF_1:def 1;
          hence contradiction by A65,AFF_1:6;
        end;
        assume LIN a,b,x;
        then LIN x,b,a by AFF_1:6;
        then
A78:    x,b // x,a by AFF_1:def 1;
        x<>b by A67,A51,A59,A56,A69,AFF_1:21;
        hence contradiction by A67,A51,A61,A68,A66,A69,A70,A78,A77,AFF_1:38;
      end;
A79:  P // C by A33,A34,AFF_1:44;
A80:  x<>c9
      proof
A81:    now
A82:      P // P by A33,AFF_1:44;
          assume
A83:      P=N;
          then c in P by A39,A40,A79,A63,AFF_1:45;
          hence contradiction by A37,A38,A52,A63,A83,A82,AFF_1:39;
        end;
        assume x=c9;
        then M=N by A63,A62,A70,AFF_1:45;
        then
A84:    P=N or b=b9 by A37,A38,A42,A60,A54,A61,AFF_1:18;
        then b,a // b,a9 by A46,A81,AFF_1:4;
        then LIN b,a,a9 by AFF_1:def 1;
        then LIN a,a9,b by AFF_1:6;
        then b in A or a=a9 by A35,A36,A41,AFF_1:25;
        then LIN a9,c,c9 by A33,A37,A44,A47,AFF_1:45,def 1;
        then LIN c,c9,a9 by AFF_1:6;
        then c =c9 or a9 in C by A39,A40,A43,AFF_1:25;
        hence contradiction by A34,A36,A45,A52,A84,A81,AFF_1:2,45;
      end;
      then D is being_line by AFF_1:24;
      then A // D by A32,A33,A35,A36,A37,A38,A41,A42,A44,A46,A71,A74,A73,A75
,A80,A76,A72;
      then D // C by A34,AFF_1:44;
      then C=D by A40,A75,AFF_1:45;
      then C=K by A39,A43,A52,A67,A59,A69,A71,A73,AFF_1:18;
      hence contradiction by A34,A35,A45,A51,AFF_1:45;
end;

theorem
  AP is satisfying_pap iff AP is satisfying_pap_1
proof
    hereby assume
A1: AP is satisfying_pap;
    thus AP is satisfying_pap_1
    proof
      let M,N,a,b,c,a9,b9,c9;
      assume that
A2:   M is being_line and
A3:   N is being_line and
A4:   a in M and
A5:   b in M and
A6:   c in M and
A7:   M // N and
A8:   M<>N and
A9:  a9 in N and
A10:  b9 in N and
A11:  a,b9 // b,a9 and
A12:  b,c9 // c,b9 and
A13:  a,c9 // c,a9 and
A14:  a9<>b9;
A15:  c <>a9 by A6,A7,A8,A9,AFF_1:45;
      set C=Line(c,b9);
A16:  c <>b9 by A6,A7,A8,A10,AFF_1:45;
      then C is being_line by AFF_1:24;
      then consider K such that
A17:  b in K and
A18:  C // K by AFF_1:49;
A19:  c in C & b9 in C by A16,AFF_1:24;
A20:  not K // N
      proof
        assume K // N;
        then C // N by A18,AFF_1:44;
        then C // M by A7,AFF_1:44;
        then b9 in M by A6,A19,AFF_1:45;
        hence contradiction by A7,A8,A10,AFF_1:45;
      end;
      K is being_line by A18,AFF_1:36;
      then consider x such that
A21:  x in K and
A22:  x in N by A3,A20,AFF_1:58;
A23:  b,x // c,b9 by A19,A17,A18,A21,AFF_1:39;
      then a,x // c,a9 by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A22;
      then a,x // a,c9 by A13,A15,AFF_1:5;
      then LIN a,x,c9 by AFF_1:def 1;
      then
A24:  LIN c9,x,a by AFF_1:6;
A25:  a<>b
      proof
        assume a=b;
        then LIN a,b9,a9 by A11,AFF_1:def 1;
        then LIN a9,b9,a by AFF_1:6;
        then a9=b9 or a in N by A3,A9,A10,AFF_1:25;
        hence contradiction by A4,A7,A8,A14,AFF_1:45;
      end;
A26:  c9<>b
      proof
        assume c9=b;
        then a9 in M by A2,A4,A5,A6,A13,A25,AFF_1:48;
        hence contradiction by A7,A8,A9,AFF_1:45;
      end;
      b,x // b,c9 by A12,A16,A23,AFF_1:5;
      then LIN b,x,c9 by AFF_1:def 1;
      then
A27:  LIN c9,x,b by AFF_1:6;
      assume
A28:  not c9 in N;
      LIN c9,x,c9 by AFF_1:7;
      then c9 in M by A2,A4,A5,A28,A25,A22,A24,A27,AFF_1:8,25;
      then b9 in M by A2,A5,A6,A12,A26,AFF_1:48;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
  end;
    assume
A29: AP is satisfying_pap_1;
      let M,N,a,b,c,a9,b9,c9;
      assume that
A30:  M is being_line and
A31:  N is being_line and
A32:  a in M and
A33:  b in M and
A34:  c in M and
A35:  M // N & M<>N and
A36:  a9 in N and
A37:  b9 in N and
A38:  c9 in N and
A39:  a,b9 // b,a9 and
A40:  b,c9 // c,b9;
      set A=Line(c,a9), D=Line(b,c9);
A41:  b<>c9 by A33,A35,A38,AFF_1:45;
      then
A42:  b in D & c9 in D by AFF_1:24;
      assume
A43:  not a,c9 // c,a9;
      then
A44:  c <>a9 by AFF_1:3;
      then
A45:  c in A by AFF_1:24;
A46:  a9 in A by A44,AFF_1:24;
A47:  A is being_line by A44,AFF_1:24;
      then consider P such that
A48:  a in P and
A49:  A // P by AFF_1:49;
A50:  a9<>b9
      proof
        assume
A51:    a9=b9;
        then a9,a // a9,b by A39,AFF_1:4;
        then LIN a9,a,b by AFF_1:def 1;
        then LIN a,b,a9 by AFF_1:6;
        then a=b or a9 in M by A30,A32,A33,AFF_1:25;
        hence contradiction by A35,A36,A40,A43,A51,AFF_1:45;
      end;
A52:  not D // P
      proof
        assume D // P;
        then c,b9 // P by A40,A41,A42,AFF_1:32,40;
        then c,b9 // A by A49,AFF_1:43;
        then b9 in A by A47,A45,AFF_1:23;
        then c in N by A31,A36,A37,A50,A47,A45,A46,AFF_1:18;
        hence contradiction by A34,A35,AFF_1:45;
      end;
A53:  D is being_line by A41,AFF_1:24;
      P is being_line by A49,AFF_1:36;
      then consider x such that
A54:  x in D and
A55:  x in P by A53,A52,AFF_1:58;
      LIN b,x,c9 by A53,A42,A54,AFF_1:21;
      then b,x // b,c9 by AFF_1:def 1;
      then
A56:  b,x // c,b9 by A40,A41,AFF_1:5;
      a,x // c,a9 by A45,A46,A48,A49,A55,AFF_1:39;
      then x in N by A29,A30,A31,A32,A33,A34,A35,A36,A37,A39,A50,A56;
      then x=c9 or b in N by A31,A38,A53,A42,A54,AFF_1:18;
      hence contradiction by A33,A35,A43,A45,A46,A48,A49,A55,AFF_1:39,45;
end;

theorem Th9:
  AP is Pappian implies AP is satisfying_pap
proof
  assume
A1: AP is Pappian;
    let M,N,a,b,c,a9,b9,c9;
    assume that
A2: M is being_line and
A3: N is being_line and
A4: a in M and
A5: b in M and
A6: c in M and
A7: M // N and
A8: M<>N and
A9: a9 in N and
A10: b9 in N and
A11: c9 in N and
A12: a,b9 // b,a9 and
A13: b,c9 // c,b9;
A14: b<>a9 by A5,A7,A8,A9,AFF_1:45;
    set K=Line(a,c9), C=Line(c,b9);
A15: c <>b9 by A6,A7,A8,A10,AFF_1:45;
    then
A16: C is being_line by AFF_1:24;
    assume
A17: not a,c9 // c,a9;
A18: now
      assume
A19:  a=b;
      then LIN a,b9,a9 by A12,AFF_1:def 1;
      then LIN a9,b9,a by AFF_1:6;
      then a9=b9 or a in N by A3,A9,A10,AFF_1:25;
      hence contradiction by A4,A7,A8,A13,A17,A19,AFF_1:45;
    end;
A20: now
      assume a9=b9;
      then a9,a // a9,b by A12,AFF_1:4;
      then LIN a9,a,b by AFF_1:def 1;
      then LIN a,b,a9 by AFF_1:6;
      then a9 in M by A2,A4,A5,A18,AFF_1:25;
      hence contradiction by A7,A8,A9,AFF_1:45;
    end;
A21: now
      assume
A22:  b=c;
      then LIN b,c9,b9 by A13,AFF_1:def 1;
      then LIN b9,c9,b by AFF_1:6;
      then b9=c9 or b in N by A3,A10,A11,AFF_1:25;
      hence contradiction by A5,A7,A8,A12,A17,A22,AFF_1:45;
    end;
A23: now
      assume b9=c9;
      then b9,b // b9,c by A13,AFF_1:4;
      then LIN b9,b,c by AFF_1:def 1;
      then LIN b,c,b9 by AFF_1:6;
      then b9 in M by A2,A5,A6,A21,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
A24: a<>c9 by A4,A7,A8,A11,AFF_1:45;
    then
A25: a in K by AFF_1:24;
    K is being_line by A24,AFF_1:24;
    then consider T such that
A26: a9 in T and
A27: K // T by AFF_1:49;
A28: T is being_line by A27,AFF_1:36;
A29: c9 in K by A24,AFF_1:24;
A30: c in C & b9 in C by A15,AFF_1:24;
    not C // T
    proof
      assume C // T;
      then C // K by A27,AFF_1:44;
      then c,b9 // a,c9 by A25,A29,A30,AFF_1:39;
      then b,c9 // a,c9 by A13,A15,AFF_1:5;
      then c9,b // c9,a by AFF_1:4;
      then LIN c9,b,a by AFF_1:def 1;
      then LIN a,b,c9 by AFF_1:6;
      then c9 in M by A2,A4,A5,A18,AFF_1:25;
      hence contradiction by A7,A8,A11,AFF_1:45;
    end;
    then consider x such that
A31: x in C and
A32: x in T by A16,A28,AFF_1:58;
    set D=Line(x,b);
A33: x<>b
    proof
      assume x=b;
      then LIN b,c,b9 by A16,A30,A31,AFF_1:21;
      then b9 in M by A2,A5,A6,A21,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
    then
A34: b in D by AFF_1:24;
    then
A35: D<>N by A5,A7,A8,AFF_1:45;
A36: D is being_line by A33,AFF_1:24;
A37: x in D by A33,AFF_1:24;
    not D // N
    proof
      assume D // N;
      then D // M by A7,AFF_1:44;
      then x in M by A5,A37,A34,AFF_1:45;
      then c in T or b9 in M by A2,A6,A16,A30,A31,A32,AFF_1:18;
      hence contradiction by A7,A8,A10,A17,A25,A29,A26,A27,AFF_1:39,45;
    end;
    then consider o such that
A38: o in D and
A39: o in N by A3,A36,AFF_1:58;
    LIN b9,c,x by A16,A30,A31,AFF_1:21;
    then
A40: b9,c // b9,x by AFF_1:def 1;
A41: a9<>x
    proof
      assume a9=x;
      then b9,a9 // b9,c by A40,AFF_1:4;
      then LIN b9,a9,c by AFF_1:def 1;
      then c in N by A3,A9,A10,A20,AFF_1:25;
      hence contradiction by A6,A7,A8,AFF_1:45;
    end;
A42: now
      assume o=x;
      then N=T by A3,A9,A26,A28,A32,A39,A41,AFF_1:18;
      then N=K by A11,A29,A27,AFF_1:45;
      hence contradiction by A4,A7,A8,A25,AFF_1:45;
    end;
A43: a,c9 // x,a9 by A25,A29,A26,A27,A32,AFF_1:39;
A44: now
      assume o=a9;
      then LIN a9,b,x by A36,A37,A34,A38,AFF_1:21;
      then a9,b // a9,x by AFF_1:def 1;
      then b,a9 // x,a9 by AFF_1:4;
      then a,b9 // x,a9 by A12,A14,AFF_1:5;
      then a,b9 // a,c9 by A43,A41,AFF_1:5;
      then LIN a,b9,c9 by AFF_1:def 1;
      then LIN b9,c9,a by AFF_1:6;
      then a in N by A3,A10,A11,A23,AFF_1:25;
      hence contradiction by A4,A7,A8,AFF_1:45;
    end;
    c,b9 // x,b9 by A40,AFF_1:4;
    then
A45: b,c9 // x,b9 by A13,A15,AFF_1:5;
A46: a<>b9 by A4,A7,A8,A10,AFF_1:45;
    not a,b9 // D
    proof
      assume a,b9 // D;
      then b,a9 // D by A12,A46,AFF_1:32;
      then a9 in D by A36,A34,AFF_1:23;
      then b in T by A26,A28,A32,A36,A37,A34,A41,AFF_1:18;
      then b,a9 // a,c9 by A25,A29,A26,A27,AFF_1:39;
      then a,b9 // a,c9 by A12,A14,AFF_1:5;
      then LIN a,b9,c9 by AFF_1:def 1;
      then LIN b9,c9,a by AFF_1:6;
      then a in N by A3,A10,A11,A23,AFF_1:25;
      hence contradiction by A4,A7,A8,AFF_1:45;
    end;
    then consider y such that
A47: y in D and
A48: LIN a,b9,y by A36,AFF_1:59;
A49: LIN a,y,a by AFF_1:7;
A50: b9<>x
    proof
      assume b9=x;
      then a,c9 // a9,b9 by A25,A29,A26,A27,A32,AFF_1:39;
      then a,c9 // N by A3,A9,A10,A20,AFF_1:27;
      then c9,a // N by AFF_1:34;
      then a in N by A3,A11,AFF_1:23;
      hence contradiction by A4,A7,A8,AFF_1:45;
    end;
A51: now
      assume o=b9;
      then LIN b9,x,b by A36,A37,A34,A38,AFF_1:21;
      then b9,x // b9,b by AFF_1:def 1;
      then x,b9 // b,b9 by AFF_1:4;
      then b,c9 // b,b9 by A45,A50,AFF_1:5;
      then LIN b,c9,b9 by AFF_1:def 1;
      then LIN b9,c9,b by AFF_1:6;
      then b in N by A3,A10,A11,A23,AFF_1:25;
      hence contradiction by A5,A7,A8,AFF_1:45;
    end;
A52: now
      assume o=y;
      then LIN b9,o,a by A48,AFF_1:6;
      then a in N by A3,A10,A39,A51,AFF_1:25;
      hence contradiction by A4,A7,A8,AFF_1:45;
    end;
A53: b<>c9 by A5,A7,A8,A11,AFF_1:45;
A54: now
      assume o=c9;
      then D // C by A13,A15,A53,A16,A30,A36,A34,A38,AFF_1:38;
      then b in C by A31,A37,A34,AFF_1:45;
      then LIN c,b,b9 by A16,A30,AFF_1:21;
      then b9 in M by A2,A5,A6,A21,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
    LIN b9,a,y by A48,AFF_1:6;
    then b9,a // b9,y by AFF_1:def 1;
    then a,b9 // y,b9 by AFF_1:4;
    then
A55: y,b9 // b,a9 by A12,A46,AFF_1:5;
    o<>b by A5,A7,A8,A39,AFF_1:45;
    then y,c9 // x,a9 by A1,A3,A9,A10,A11,A36,A37,A34,A38,A39,A45,A47,A55,A35
,A51,A44,A54,A42,A52;
    then y,c9 // a,c9 by A43,A41,AFF_1:5;
    then c9,y // c9,a by AFF_1:4;
    then LIN c9,y,a by AFF_1:def 1;
    then
A56: LIN a,y,c9 by AFF_1:6;
    LIN a,y,b9 by A48,AFF_1:6;
    then a in D or a in N by A3,A10,A11,A23,A47,A56,A49,AFF_1:8,25;
    then D // N by A2,A4,A5,A7,A8,A18,A36,A34,AFF_1:18,45;
    hence contradiction by A38,A39,A35,AFF_1:45;
end;

theorem Th10:
  AP is satisfying_PPAP iff AP is Pappian & AP is satisfying_pap
proof
A1: AP is Pappian & AP is satisfying_pap implies AP is satisfying_PPAP
  proof
    assume that
A2: AP is Pappian and
A3: AP is satisfying_pap;
    thus AP is satisfying_PPAP
    proof
      let M,N,a,b,c,a9,b9,c9;
      assume that
A4:   M is being_line and
A5:   N is being_line and
A6:   a in M and
A7:   b in M and
A8:   c in M and
A9:   a9 in N and
A10:  b9 in N and
A11:  c9 in N and
A12:  a,b9 // b,a9 and
A13:  b,c9 // c,b9;
      now
        assume
A14:    M<>N;
        assume
A15:    not thesis;
        now
          assume not M // N;
          then consider o such that
A16:      o in M and
A17:      o in N by A4,A5,AFF_1:58;
A18:      o<>a
          proof
            assume
A19:        o=a;
            then o,b9 // a9,b by A12,AFF_1:4;
            then o=b9 or b in N by A5,A9,A10,A17,AFF_1:48;
            then o=b9 or o=b by A4,A5,A7,A14,A16,A17,AFF_1:18;
            then c,o // b,c9 or o,c9 // b9,c by A13,AFF_1:4;
            then c9 in M or c =o or c in N or o=c9 by A4,A5,A7,A8,A10,A11,A16
,A17,AFF_1:48;
            then o=c or o=c9 by A4,A5,A8,A11,A14,A16,A17,AFF_1:18;
            hence contradiction by A5,A9,A11,A15,A17,A19,AFF_1:3,51;
          end;
A20:      o<>b
          proof
            assume
A21:        o=b;
            then o,c9 // b9,c by A13,AFF_1:4;
            then o=c9 or c in N by A5,A10,A11,A17,AFF_1:48;
            then
A22:        o=c9 or o=c by A4,A5,A8,A14,A16,A17,AFF_1:18;
            o,a9 // b9,a by A12,A21,AFF_1:4;
            then o=a9 or a in N by A5,A9,A10,A17,AFF_1:48;
            hence contradiction by A4,A5,A6,A8,A14,A15,A16,A17,A18,A22,AFF_1:3
,18,51;
          end;
A23:      o<>c9
          proof
            assume
A24:        o=c9;
            then b9 in M by A4,A7,A8,A13,A16,A20,AFF_1:48;
            then a,o // b,a9 by A4,A5,A10,A12,A14,A16,A17,AFF_1:18;
            then a9 in M by A4,A6,A7,A16,A18,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A16,A24,AFF_1:51;
          end;
A25:      o<>c
          proof
            assume
A26:        o=c;
            then o,b9 // c9,b by A13,AFF_1:4;
            then o=b9 or b in N by A5,A10,A11,A17,AFF_1:48;
            then a9 in M by A4,A5,A6,A7,A9,A12,A16,A17,A18,A20,AFF_1:18,48;
            then a9=o by A4,A5,A9,A14,A16,A17,AFF_1:18;
            hence contradiction by A15,A26,AFF_1:3;
          end;
A27:      o<>a9
          proof
            assume
A28:        o=a9;
            then o,b // a,b9 by A12,AFF_1:4;
            then b9 in M by A4,A6,A7,A16,A20,AFF_1:48;
            then o=b9 by A4,A5,A10,A14,A16,A17,AFF_1:18;
            then c,o // b,c9 by A13,AFF_1:4;
            then c9 in M by A4,A7,A8,A16,A25,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A16,A28,AFF_1:51;
          end;
          o<>b9
          proof
            assume
A29:        o=b9;
            then o,c // b,c9 by A13,AFF_1:4;
            then
A30:        c9 in M by A4,A7,A8,A16,A25,AFF_1:48;
            a9 in M by A4,A6,A7,A12,A16,A18,A29,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A30,AFF_1:51;
          end;
          hence thesis by A2,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A16,A17,A18
,A20,A25,A27,A23;
        end;
        hence thesis by A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14;
      end;
      hence thesis by A4,A6,A8,A9,A11,AFF_1:51;
    end;
  end;
  thus thesis by A1;
end;

theorem
  AP is Pappian implies AP is Desarguesian
proof
  assume
A1: AP is Pappian;
  then AP is satisfying_pap by Th9;
  then
A2: AP is satisfying_PPAP by A1,Th10;
    let A,P,C,o,a,b,c,a9,b9,c9;
    assume that
A3: o in A and
A4: o in P and
A5: o in C and
A6: o<>a and
A7: o<>b and
    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;
    assume
A21: not b,c // b9,c9;
    then
A22: b<>c by AFF_1:3;
A23: a<>c by A3,A5,A6,A8,A12,A14,A16,A18,AFF_1:18;
A24: not b in C
    proof
      assume
A25:  b in C;
      then b9 in C by A4,A5,A7,A10,A11,A15,A16,AFF_1:18;
      hence contradiction by A12,A13,A16,A21,A25,AFF_1:51;
    end;
A26: a<>b by A3,A4,A6,A8,A10,A14,A15,A17,AFF_1:18;
A27: a<>a9
    proof
      assume
A28:  a=a9;
      then LIN a,c,c9 by A20,AFF_1:def 1;
      then LIN c,c9,a by AFF_1:6;
      then
A29:  c =c9 or a in C by A12,A13,A16,AFF_1:25;
      LIN a,b,b9 by A19,A28,AFF_1:def 1;
      then LIN b,b9,a by AFF_1:6;
      then b=b9 or a in P by A10,A11,A15,AFF_1:25;
      hence contradiction by A3,A4,A5,A6,A8,A14,A15,A16,A17,A18,A21,A29,AFF_1:2
,18;
    end;
    set M=Line(b9,c9), N=Line(a9,b9), D=Line(a9,c9);
A30: a9<>b9
    proof
A31:  a9,c9 // c,a by A20,AFF_1:4;
      assume
A32:  a9=b9;
      then a9 in C by A3,A4,A5,A9,A11,A14,A15,A17,AFF_1:18;
      then a in C or a9=c9 by A12,A13,A16,A31,AFF_1:48;
      hence contradiction by A3,A5,A6,A8,A14,A16,A18,A21,A32,AFF_1:3,18;
    end;
    then
A33: N is being_line by AFF_1:24;
A34: a9<>c9
    proof
      assume a9=c9;
      then
A35:  a9 in P by A3,A4,A5,A9,A13,A14,A16,A18,AFF_1:18;
      a9,b9 // b,a by A19,AFF_1:4;
      then a in P by A10,A11,A15,A30,A35,AFF_1:48;
      hence contradiction by A3,A4,A6,A8,A14,A15,A17,AFF_1:18;
    end;
A36: not LIN a9,b9,c9
    proof
      assume
A37:  LIN a9,b9,c9;
      then a9,b9 // a9,c9 by AFF_1:def 1;
      then a9,b9 // a,c by A20,A34,AFF_1:5;
      then a,b // a,c by A19,A30,AFF_1:5;
      then LIN a,b,c by AFF_1:def 1;
      then LIN b,c,a by AFF_1:6;
      then b,c // b,a by AFF_1:def 1;
      then b,c // a,b by AFF_1:4;
      then
A38:  b,c // a9,b9 by A19,A26,AFF_1:5;
      LIN b9,c9,a9 by A37,AFF_1:6;
      then b9,c9 // b9,a9 by AFF_1:def 1;
      then b9,c9 // a9,b9 by AFF_1:4;
      hence contradiction by A21,A30,A38,AFF_1:5;
    end;
A39: not LIN a,b,c
    proof
      assume LIN a,b,c;
      then a,b // a,c by AFF_1:def 1;
      then a,b // a9,c9 by A20,A23,AFF_1:5;
      then a9,b9 // a9,c9 by A19,A26,AFF_1:5;
      hence contradiction by A36,AFF_1:def 1;
    end;
A40: now
      LIN o,a,a9 by A3,A8,A9,A14,AFF_1:21;
      then o,a // o,a9 by AFF_1:def 1;
      then
A41:  a9,o // a,o by AFF_1:4;
      set M=Line(b,c), N=Line(a,b), D=Line(a,c);
A42:  N is being_line by A26,AFF_1:24;
      M is being_line by A22,AFF_1:24;
      then consider K such that
A43:  o in K and
A44:  M // K by AFF_1:49;
A45:  K is being_line by A44,AFF_1:36;
A46:  a in N by A26,AFF_1:24;
A47:  b in N by A26,AFF_1:24;
A48:  b in M & c in M by A22,AFF_1:24;
      not N // K
      proof
        assume N // K;
        then N // M by A44,AFF_1:44;
        then c in N by A48,A47,AFF_1:45;
        hence contradiction by A39,A42,A46,A47,AFF_1:21;
      end;
      then consider p such that
A49:  p in N and
A50:  p in K by A42,A45,AFF_1:58;
A51:  b,c // p,o by A48,A43,A44,A50,AFF_1:39;
A52:  o<>p
      proof
        assume o=p;
        then LIN o,a,b by A42,A46,A47,A49,AFF_1:21;
        then b in A by A3,A6,A8,A14,AFF_1:25;
        hence contradiction by A3,A4,A7,A10,A14,A15,A17,AFF_1:18;
      end;
      set R=Line(a9,p);
A53:  p<>a9
      proof
        assume p=a9;
        then b in A by A8,A9,A14,A27,A42,A46,A47,A49,AFF_1:18;
        hence contradiction by A3,A4,A7,A10,A14,A15,A17,AFF_1:18;
      end;
      then
A54:  R is being_line by AFF_1:24;
      D is being_line by A23,AFF_1:24;
      then consider T such that
A55:  p in T and
A56:  D // T by AFF_1:49;
A57:  a in D & c in D by A23,AFF_1:24;
A58:  not C // T
      proof
        assume C // T;
        then C // D by A56,AFF_1:44;
        then a in C by A12,A57,AFF_1:45;
        hence contradiction by A3,A5,A6,A8,A14,A16,A18,AFF_1:18;
      end;
      T is being_line by A56,AFF_1:36;
      then consider q such that
A59:  q in C and
A60:  q in T by A16,A58,AFF_1:58;
A61:  p,q // a,c by A57,A55,A56,A60,AFF_1:39;
      then
A62:  b,q // a,o by A2,A5,A12,A16,A42,A46,A47,A49,A59,A51;
A63:  a9 in R & p in R by A53,AFF_1:24;
      assume not b,c // A;
      then
A64:  K<>A by A48,A44,AFF_1:40;
      not b,q // R
      proof
        assume b,q // R;
        then
A65:    a,o // R by A24,A59,A62,AFF_1:32;
        a,o // A by A3,A8,A14,AFF_1:40,41;
        then p in A by A6,A9,A63,A65,AFF_1:45,53;
        hence contradiction by A3,A14,A43,A45,A50,A52,A64,AFF_1:18;
      end;
      then consider r such that
A66:  r in R and
A67:  LIN b,q,r by A54,AFF_1:59;
A68:  now
        assume r=q;
        then b,r // a,o by A2,A5,A12,A16,A42,A46,A47,A49,A59,A51,A61;
        then
A69:    r,b // o,a by AFF_1:4;
        LIN o,a,a9 by A3,A8,A9,A14,AFF_1:21;
        then o,a // o,a9 by AFF_1:def 1;
        hence r,b // o,a9 by A6,A69,AFF_1:5;
      end;
      LIN q,r,b by A67,AFF_1:6;
      then q,r // q,b by AFF_1:def 1;
      then r,q // b,q by AFF_1:4;
      then r,q // a,o by A24,A59,A62,AFF_1:5;
      then
A70:  a9,o // r,q by A6,A41,AFF_1:5;
      LIN b,a,p by A42,A46,A47,A49,AFF_1:21;
      then b,a // b,p by AFF_1:def 1;
      then a,b // p,b by AFF_1:4;
      then
A71:  p,b // a9,b9 by A19,A26,AFF_1:5;
      LIN r,b,q by A67,AFF_1:6;
      then r,b // r,q by AFF_1:def 1;
      then a9,o // r,b by A70,A68,AFF_1:4,5;
      then
A72:  p,o // r,b9 by A2,A4,A10,A11,A15,A54,A63,A66,A71;
      p,q // a9,c9 by A20,A23,A61,AFF_1:5;
      then
A73:  p,o // r,c9 by A2,A5,A13,A16,A59,A54,A63,A66,A70;
      then r,c9 // r,b9 by A52,A72,AFF_1:5;
      then LIN r,c9,b9 by AFF_1:def 1;
      then LIN c9,b9,r by AFF_1:6;
      then c9,b9 // c9,r by AFF_1:def 1;
      then
A74:  r,c9 // b9,c9 by AFF_1:4;
      b,c // r,c9 by A52,A51,A73,AFF_1:5;
      then r=c9 by A21,A74,AFF_1:5;
      then p,o // b9,c9 by A72,AFF_1:4;
      hence contradiction by A21,A52,A51,AFF_1:5;
    end;
A75: b9 in N by A30,AFF_1:24;
A76: b9<>c9 by A21,AFF_1:3;
    then
A77: b9 in M & c9 in M by AFF_1:24;
    M is being_line by A76,AFF_1:24;
    then consider K such that
A78: o in K and
A79: M // K by AFF_1:49;
A80: K is being_line by A79,AFF_1:36;
A81: a9 in N by A30,AFF_1:24;
    not N // K
    proof
      assume N // K;
      then N // M by A79,AFF_1:44;
      then c9 in N by A77,A75,AFF_1:45;
      hence contradiction by A36,A33,A81,A75,AFF_1:21;
    end;
    then consider p such that
A82: p in N and
A83: p in K by A33,A80,AFF_1:58;
A84: o<>a9
    proof
      assume
A85:  o=a9;
      a9,b9 // b,a by A19,AFF_1:4;
      then a in P by A4,A10,A11,A15,A30,A85,AFF_1:48;
      hence contradiction by A3,A4,A6,A8,A14,A15,A17,AFF_1:18;
    end;
A86: o<>p
    proof
      assume o=p;
      then LIN o,a9,b9 by A33,A81,A75,A82,AFF_1:21;
      then
A87:  b9 in A by A3,A9,A14,A84,AFF_1:25;
      a9,b9 // a,b by A19,AFF_1:4;
      then b in A by A8,A9,A14,A30,A87,AFF_1:48;
      hence contradiction by A3,A4,A7,A10,A14,A15,A17,AFF_1:18;
    end;
    D is being_line by A34,AFF_1:24;
    then consider T such that
A88: p in T and
A89: D // T by AFF_1:49;
A90: T is being_line by A89,AFF_1:36;
A91: a9 in D & c9 in D by A34,AFF_1:24;
    not C // T
    proof
      assume C // T;
      then C // D by A89,AFF_1:44;
      then a9 in C by A13,A91,AFF_1:45;
      hence contradiction by A3,A5,A9,A14,A16,A18,A84,AFF_1:18;
    end;
    then consider q such that
A92: q in C and
A93: q in T by A16,A90,AFF_1:58;
A94: b9,c9 // p,o by A77,A78,A79,A83,AFF_1:39;
A95: o<>b9
    proof
      assume
A96:  o=b9;
      b9,a9 // a,b by A19,AFF_1:4;
      then b in A by A3,A8,A9,A14,A30,A96,AFF_1:48;
      hence contradiction by A3,A4,A7,A10,A14,A15,A17,AFF_1:18;
    end;
A97: b9<>q
    proof
      assume b9=q;
      then P=C by A4,A5,A11,A15,A16,A95,A92,AFF_1:18;
      hence contradiction by A10,A11,A12,A13,A15,A21,AFF_1:51;
    end;
    set R=Line(a,p);
A98: p<>a
    proof
      assume p=a;
      then b9 in A by A8,A9,A14,A27,A33,A81,A75,A82,AFF_1:18;
      hence contradiction by A3,A4,A11,A14,A15,A17,A95,AFF_1:18;
    end;
    then
A99: R is being_line by AFF_1:24;
A100: p,q // a9,c9 by A91,A88,A89,A93,AFF_1:39;
    then
A101: b9,q // a9,o by A2,A5,A13,A16,A33,A81,A75,A82,A92,A94;
A102: a in R & p in R by A98,AFF_1:24;
    not b9,c9 // A by A14,A21,A40,AFF_1:31;
    then
A103: K<>A by A77,A79,AFF_1:40;
    not b9,q // R
    proof
      assume b9,q // R;
      then
A104: a9,o // R by A101,A97,AFF_1:32;
      a9,o // A by A3,A9,A14,AFF_1:40,41;
      then p in A by A8,A84,A102,A104,AFF_1:45,53;
      hence contradiction by A3,A14,A78,A80,A83,A86,A103,AFF_1:18;
    end;
    then consider r such that
A105: r in R and
A106: LIN b9,q,r by A99,AFF_1:59;
A107: now
      assume r=q;
      then b9,r // a9,o by A2,A5,A13,A16,A33,A81,A75,A82,A92,A94,A100;
      then
A108: r,b9 // o,a9 by AFF_1:4;
      LIN o,a9,a by A3,A8,A9,A14,AFF_1:21;
      then o,a9 // o,a by AFF_1:def 1;
      hence r,b9 // o,a by A84,A108,AFF_1:5;
    end;
    LIN b9,a9,p by A33,A81,A75,A82,AFF_1:21;
    then b9,a9 // b9,p by AFF_1:def 1;
    then p,b9 // a9,b9 by AFF_1:4;
    then
A109: p,b9 // a,b by A19,A30,AFF_1:5;
    LIN o,a,a9 by A3,A8,A9,A14,AFF_1:21;
    then o,a // o,a9 by AFF_1:def 1;
    then
A110: a,o // a9,o by AFF_1:4;
    LIN q,r,b9 by A106,AFF_1:6;
    then q,r // q,b9 by AFF_1:def 1;
    then r,q // b9,q by AFF_1:4;
    then r,q // a9,o by A101,A97,AFF_1:5;
    then
A111: a,o // r,q by A84,A110,AFF_1:5;
    LIN r,b9,q by A106,AFF_1:6;
    then r,b9 // r,q by AFF_1:def 1;
    then a,o // r,b9 by A111,A107,AFF_1:4,5;
    then
A112: p,o // r,b by A2,A4,A10,A11,A15,A99,A102,A105,A109;
    p,q // a,c by A20,A34,A100,AFF_1:5;
    then
A113: p,o // r,c by A2,A5,A12,A16,A92,A99,A102,A105,A111;
    then r,c // r,b by A86,A112,AFF_1:5;
    then LIN r,c,b by AFF_1:def 1;
    then LIN c,b,r by AFF_1:6;
    then c,b // c,r by AFF_1:def 1;
    then
A114: b,c // r,c by AFF_1:4;
    b9,c9 // r,c by A86,A94,A113,AFF_1:5;
    then r=c by A21,A114,AFF_1:5;
    then b,c // p,o by A112,AFF_1:4;
    hence contradiction by A21,A86,A94,AFF_1:5;
end;

theorem
  AP is Desarguesian implies AP is Moufangian
proof
  assume
A1: AP is Desarguesian;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K & c9 in K and
A5: not a in K and
A6: o<>c and
A7: a<>b and
A8: LIN o,a,a9 and
A9: LIN o,b,b9 and
A10: a,b // a9,b9 & a,c // a9,c9 and
A11: a,b // K;
    set A=Line(o,a), P=Line(o,b);
A12: o in A by A3,A5,AFF_1:24;
A13: now
      assume
A14:  o=b;
      b,a // K by A11,AFF_1:34;
      hence contradiction by A2,A3,A5,A14,AFF_1:23;
    end;
    then
A15: b in P by AFF_1:24;
A16: a in A by A3,A5,AFF_1:24;
A17: A is being_line by A3,A5,AFF_1:24;
A18: A<>P
    proof
      assume A=P;
      then a,b // A by A17,A16,A15,AFF_1:40,41;
      hence contradiction by A3,A5,A7,A11,A12,A16,AFF_1:45,53;
    end;
A19: P is being_line & o in P by A13,AFF_1:24;
    then
A20: b9 in P by A9,A13,A15,AFF_1:25;
    a9 in A by A3,A5,A8,A17,A12,A16,AFF_1:25;
    hence thesis by A1,A2,A3,A4,A5,A6,A10,A13,A17,A12,A16,A19,A15,A20,A18;
end;

theorem Th13:
  AP is satisfying_TDES_1 implies AP is satisfying_des_1
proof
  assume
A1: AP is satisfying_TDES_1;
    let A,P,C,a,b,c,a9,b9,c9;
    assume that
A2: A // P and
A3: a in A and
A4: a9 in A and
A5: b in P and
A6: b9 in P and
A7: c in C and
A8: c9 in C and
A9: A is being_line and
A10: P is being_line and
A11: C is being_line and
A12: A<>P and
A13: A<>C and
A14: a,b // a9,b9 and
A15: a,c // a9,c9 and
A16: b,c // b9,c9 and
A17: not LIN a,b,c and
A18: c <>c9;
    set P9=Line(a,b), C9=Line(a9,b9);
A19: a9<>b9 by A2,A4,A6,A12,AFF_1:45;
    then
A20: a9 in C9 by AFF_1:24;
A21: a9<>c9
    proof
      assume a9=c9;
      then b,c // a9,b9 by A16,AFF_1:4;
      then a,b // b,c by A14,A19,AFF_1:5;
      then b,a // b,c by AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      hence contradiction by A17,AFF_1:6;
    end;
A22: not c9 in A
    proof
      assume
A23:  c9 in A;
      a9,c9 // a,c by A15,AFF_1:4;
      then c in A by A3,A4,A9,A21,A23,AFF_1:48;
      hence contradiction by A7,A8,A9,A11,A13,A18,A23,AFF_1:18;
    end;
A24: C9 is being_line by A19,AFF_1:24;
    assume
A25: not A // C;
    then consider o such that
A26: o in A and
A27: o in C by A9,A11,AFF_1:58;
A28: LIN o,c9,c by A7,A8,A11,A27,AFF_1:21;
A29: a<>a9
    proof
      assume
A30:  a=a9;
      then LIN a,c,c9 by A15,AFF_1:def 1;
      then
A31:  LIN c,c9,a by AFF_1:6;
      LIN a,b,b9 by A14,A30,AFF_1:def 1;
      then LIN b,b9,a by AFF_1:6;
      then b=b9 or a in P by A5,A6,A10,AFF_1:25;
      then LIN b,c,c9 by A2,A3,A12,A16,AFF_1:45,def 1;
      then
A32:  LIN c,c9,b by AFF_1:6;
      LIN c,c9,c by AFF_1:7;
      hence contradiction by A17,A18,A31,A32,AFF_1:8;
    end;
A33: o<>a9
    proof
      assume
A34:  o=a9;
      a9,c9 // c,a by A15,AFF_1:4;
      then a in C by A7,A8,A11,A27,A21,A34,AFF_1:48;
      hence contradiction by A3,A4,A9,A11,A13,A27,A29,A34,AFF_1:18;
    end;
A35: a<>b by A17,AFF_1:7;
    then
A36: P9 is being_line by AFF_1:24;
    consider N such that
A37: c9 in N and
A38: A // N by A9,AFF_1:49;
A39: b9 in C9 by A19,AFF_1:24;
A40: not N // C9
    proof
      assume N // C9;
      then A // C9 by A38,AFF_1:44;
      then b9 in A by A4,A39,A20,AFF_1:45;
      hence contradiction by A2,A6,A12,AFF_1:45;
    end;
    N is being_line by A38,AFF_1:36;
    then consider q such that
A41: q in N and
A42: q in C9 by A24,A40,AFF_1:58;
A43: c9,q // A by A37,A38,A41,AFF_1:40;
A44: c9<>q
    proof
      assume c9=q;
      then LIN a9,b9,c9 by A24,A39,A20,A42,AFF_1:21;
      then a9,b9 // a9,c9 by AFF_1:def 1;
      then a9,b9 // a,c by A15,A21,AFF_1:5;
      then a,b // a,c by A14,A19,AFF_1:5;
      hence contradiction by A17,AFF_1:def 1;
    end;
A45: c9,a9 // c,a by A15,AFF_1:4;
A46: c,b // c9,b9 by A16,AFF_1:4;
A47: a in P9 by A35,AFF_1:24;
A48: b<>c by A17,AFF_1:7;
A49: not c in P
    proof
      assume
A50:  c in P;
      then c9 in P by A5,A6,A10,A16,A48,AFF_1:48;
      hence contradiction by A2,A7,A8,A10,A11,A18,A25,A50,AFF_1:18;
    end;
    not P // C by A2,A25,AFF_1:44;
    then consider s such that
A51: s in P and
A52: s in C by A10,A11,AFF_1:58;
A53: LIN s,c,c9 by A7,A8,A11,A52,AFF_1:21;
A54: b<>b9
    proof
      assume b=b9;
      then b,a // b,a9 by A14,AFF_1:4;
      then LIN b,a,a9 by AFF_1:def 1;
      then LIN a,a9,b by AFF_1:6;
      then b in A by A3,A4,A9,A29,AFF_1:25;
      hence contradiction by A2,A5,A12,AFF_1:45;
    end;
A55: s<>b
    proof
      assume
A56:  s=b;
      b,c // c9,b9 by A16,AFF_1:4;
      then b9 in C by A7,A8,A11,A48,A52,A56,AFF_1:48;
      hence contradiction by A2,A5,A6,A10,A11,A25,A54,A52,A56,AFF_1:18;
    end;
    consider M such that
A57: c in M and
A58: A // M by A9,AFF_1:49;
A59: M is being_line by A58,AFF_1:36;
A60: b in P9 by A35,AFF_1:24;
    not M // P9
    proof
      assume M // P9;
      then A // P9 by A58,AFF_1:44;
      then b in A by A3,A60,A47,AFF_1:45;
      hence contradiction by A2,A5,A12,AFF_1:45;
    end;
    then consider p such that
A61: p in M and
A62: p in P9 by A59,A36,AFF_1:58;
    M // P by A2,A58,AFF_1:44;
    then
A63: c,p // P by A57,A61,AFF_1:40;
A64: M // N by A58,A38,AFF_1:44;
    then
A65: c,p // c9,q by A57,A37,A61,A41,AFF_1:39;
A66: now
      assume p=q;
      then M=N by A64,A61,A41,AFF_1:45;
      hence contradiction by A7,A8,A11,A18,A25,A57,A58,A37,A59,AFF_1:18;
    end;
A67: P9 // C9 by A14,A35,A19,AFF_1:37;
    then
A68: p,b // q,b9 by A60,A39,A62,A42,AFF_1:39;
A69: q,a9 // p,a by A47,A20,A67,A62,A42,AFF_1:39;
    c9,q // c,p by A57,A37,A64,A61,A41,AFF_1:39;
    then LIN o,q,p by A1,A3,A4,A9,A26,A28,A22,A45,A69,A43,A44,A33;
    then
A70: LIN p,q,o by AFF_1:6;
    c <>p by A17,A36,A60,A47,A62,AFF_1:21;
    then LIN s,p,q by A1,A5,A6,A10,A51,A53,A63,A68,A49,A55,A65,A46;
    then
A71: LIN p,q,s by AFF_1:6;
    LIN p,q,p by AFF_1:7;
    then o=s or p in C by A11,A27,A52,A70,A71,A66,AFF_1:8,25;
    then c in P9 by A2,A7,A11,A12,A25,A26,A57,A58,A59,A61,A62,A51,AFF_1:18,45;
    hence contradiction by A17,A36,A60,A47,AFF_1:21;
end;

theorem
  AP is Moufangian implies AP is translational
proof
  assume AP is Moufangian;
  then AP is satisfying_des_1 by Th3,Th13;
  hence thesis by Th7;
end;

theorem
  AP is translational implies AP is satisfying_pap
proof
  assume
A1: AP is translational;
    let M,N,a,b,c,a9,b9,c9;
    assume that
A2: M is being_line and
A3: N is being_line and
A4: a in M and
A5: b in M and
A6: c in M and
A7: M // N and
A8: M<>N and
A9: a9 in N and
A10: b9 in N and
A11: c9 in N and
A12: a,b9 // b,a9 and
A13: b,c9 // c,b9;
    set A=Line(a,b9), A9=Line(b,a9), P=Line(b,c9), P9=Line(c,b9);
A14: c <>b9 by A6,A7,A8,A10,AFF_1:45;
    then
A15: c in P9 & b9 in P9 by AFF_1:24;
A16: b<>c9 by A5,A7,A8,A11,AFF_1:45;
    then
A17: P is being_line & b in P by AFF_1:24;
A18: P9 is being_line by A14,AFF_1:24;
    then consider C9 such that
A19: a in C9 and
A20: P9 // C9 by AFF_1:49;
A21: C9 is being_line by A20,AFF_1:36;
    assume
A22: not thesis;
A23: now
      assume
A24:  a=c;
      then b,c9 // b,a9 by A12,A13,A14,AFF_1:5;
      then LIN b,c9,a9 by AFF_1:def 1;
      then LIN a9,c9,b by AFF_1:6;
      then a9=c9 or b in N by A3,A9,A11,AFF_1:25;
      hence contradiction by A5,A7,A8,A22,A24,AFF_1:2,45;
    end;
A25: P9<>C9
    proof
      assume P9=C9;
      then LIN a,c,b9 by A18,A15,A19,AFF_1:21;
      then b9 in M by A2,A4,A6,A23,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
A26: c9 in P by A16,AFF_1:24;
    then
A27: P9 // P by A13,A16,A14,A18,A17,A15,AFF_1:38;
A28: now
      assume
A29:  b=c;
      then LIN b,c9,b9 by A13,AFF_1:def 1;
      then LIN b9,c9,b by AFF_1:6;
      then b9=c9 or b in N by A3,A10,A11,AFF_1:25;
      hence contradiction by A5,A7,A8,A12,A22,A29,AFF_1:45;
    end;
A30: P9<>P
    proof
      assume P9=P;
      then LIN b,c,b9 by A17,A15,AFF_1:21;
      then b9 in M by A2,A5,A6,A28,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
A31: a<>b9 by A4,A7,A8,A10,AFF_1:45;
    then
A32: A is being_line by AFF_1:24;
    then consider C such that
A33: c in C and
A34: A // C by AFF_1:49;
A35: C is being_line by A34,AFF_1:36;
A36: a,b // b9,a9 by A4,A5,A7,A9,A10,AFF_1:39;
A37: b9,c9 // c,b by A5,A6,A7,A10,A11,AFF_1:39;
A38: b<>a9 by A5,A7,A8,A9,AFF_1:45;
    then
A39: a9 in A9 by AFF_1:24;
A40: a in A & b9 in A by A31,AFF_1:24;
A41: A<>C
    proof
      assume A=C;
      then LIN a,c,b9 by A32,A40,A33,AFF_1:21;
      then b9 in M by A2,A4,A6,A23,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
    not C // C9
    proof
      assume C // C9;
      then A // C9 by A34,AFF_1:44;
      then A // P9 by A20,AFF_1:44;
      then b9,a // b9,c by A40,A15,AFF_1:39;
      then LIN b9,a,c by AFF_1:def 1;
      then LIN a,c,b9 by AFF_1:6;
      then b9 in M by A2,A4,A6,A23,AFF_1:25;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
    then consider s such that
A42: s in C and
A43: s in C9 by A35,A21,AFF_1:58;
A44: A9 is being_line & b in A9 by A38,AFF_1:24;
A45: now
      assume
A46:  a=b;
      then LIN a,b9,a9 by A12,AFF_1:def 1;
      then LIN a9,b9,a by AFF_1:6;
      then a9=b9 or a in N by A3,A9,A10,AFF_1:25;
      hence contradiction by A4,A7,A8,A13,A22,A46,AFF_1:45;
    end;
A47: now
      assume a9=b9;
      then a9,a // a9,b by A12,AFF_1:4;
      then LIN a9,a,b by AFF_1:def 1;
      then LIN a,b,a9 by AFF_1:6;
      then a9 in M by A2,A4,A5,A45,AFF_1:25;
      hence contradiction by A7,A8,A9,AFF_1:45;
    end;
A48: A<>A9
    proof
      assume A=A9;
      then LIN a9,b9,a by A32,A40,A39,AFF_1:21;
      then a in N by A3,A9,A10,A47,AFF_1:25;
      hence contradiction by A4,A7,A8,AFF_1:45;
    end;
A49: b<>s
    proof
      assume b=s;
      then a,b9 // b,c by A40,A33,A34,A42,AFF_1:39;
      then b,a9 // b,c by A12,A31,AFF_1:5;
      then LIN b,a9,c by AFF_1:def 1;
      then LIN b,c,a9 by AFF_1:6;
      then a9 in M by A2,A5,A6,A28,AFF_1:25;
      hence contradiction by A7,A8,A9,AFF_1:45;
    end;
A50: A // A9 by A12,A31,A38,AFF_1:37;
    a,s // b9,c by A15,A19,A20,A43,AFF_1:39;
    then
A51: b,s // a9,c by A1,A32,A40,A44,A39,A33,A34,A35,A42,A36,A48,A41,A50;
    b9,a // c,s by A40,A33,A34,A42,AFF_1:39;
    then c9,a // b,s by A1,A18,A17,A26,A15,A19,A20,A21,A43,A37,A30,A25,A27;
    then c9,a // a9,c by A51,A49,AFF_1:5;
    hence contradiction by A22,AFF_1:4;
end;