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;

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;
