reserve AP for AffinPlane;
reserve a,a9,b,b9,c,c9,d,x,y,o,p,q for Element of AP;
reserve A,C,D9,M,N,P for Subset of AP;

theorem
  AP is satisfying_DES1_2 implies AP is Desarguesian
proof
  assume
A1: AP is satisfying_DES1_2;
  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 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
A21: not LIN o,b,a & not LIN o,a,c
    proof
A22:  now
        assume LIN o,a,c;
        then c in A by A2,A5,A8,A14,AFF_1:25;
        hence contradiction by A2,A4,A7,A12,A14,A16,A18,AFF_1:18;
      end;
A23:  now
        assume LIN o,b,a;
        then a in P by A3,A6,A10,A15,AFF_1:25;
        hence contradiction by A2,A3,A5,A8,A14,A15,A17,AFF_1:18;
      end;
      assume not thesis;
      hence thesis by A23,A22;
    end;
A24: b=b9 implies thesis
    proof
A25:  LIN o,c,c9 by A4,A12,A13,A16,AFF_1:21;
A26:  LIN o,a,a9 by A2,A8,A9,A14,AFF_1:21;
      assume
A27:  b=b9;
      then b,a // b,a9 by A19,AFF_1:4;
      then a,c // a,c9 by A20,A21,A26,AFF_1:14;
      then c =c9 by A21,A25,AFF_1:14;
      hence thesis by A27,AFF_1:2;
    end;
A28: a9=o implies thesis
    proof
      assume
A29:  a9=o;
      LIN o,b,b9 & not LIN o,a,b by A3,A10,A11,A15,A21,AFF_1:6,21;
      then
A30:  o=b9 by A19,A29,AFF_1:55;
      LIN o,c,c9 by A4,A12,A13,A16,AFF_1:21;
      then o=c9 by A20,A21,A29,AFF_1:55;
      hence thesis by A30,AFF_1:3;
    end;
A31: c9=o implies thesis
    proof
A32:  c,a // c9,a9 by A20,AFF_1:4;
      assume
A33:  c9=o;
      LIN o,a,a9 & not LIN o,c,a by A2,A8,A9,A14,A21,AFF_1:6,21;
      hence thesis by A28,A33,A32,AFF_1:55;
    end;
    set K=Line(a,c);
A34: a in K by AFF_1:15;
A35: a<>c by A2,A4,A5,A8,A12,A14,A16,A18,AFF_1:18;
    then
A36: K is being_line by AFF_1:def 3;
A37: c in K by AFF_1:15;
A38: a<>b by A2,A3,A5,A8,A10,A14,A15,A17,AFF_1:18;
A39: LIN a,b,c implies thesis
    proof
      consider N such that
A40:  a9 in N and
A41:  K // N by A36,AFF_1:49;
A42:  N is being_line by A41,AFF_1:36;
      a9,c9 // K by A20,A35,AFF_1:29,32;
      then a9,c9 // N by A41,AFF_1:43;
      then
A43:  c9 in N by A40,A42,AFF_1:23;
      assume LIN a,b,c;
      then LIN a,c,b by AFF_1:6;
      then
A44:  b in K by AFF_1:def 2;
      then K=Line(a,b) by A38,A36,A34,AFF_1:57;
      then a9,b9 // K by A19,A38,AFF_1:29,32;
      then a9,b9 // N by A41,AFF_1:43;
      then b9 in N by A40,A42,AFF_1:23;
      hence thesis by A37,A44,A41,A43,AFF_1:39;
    end;
    assume
A45: P<>C;
A46: now
      set T=Line(b9,a9);
      set D=Line(b,a);
      set N=Line(a9,c9);
      assume that
A47:  o<>a9 and
A48:  o<>b9 and
A49:  o<>c9 and
A50:  b<>b9 and
A51:  not LIN a,b,c;
A52:  c9 in N by AFF_1:15;
      assume not b,c // b9,c9;
      then consider q such that
A53:  LIN b,c,q and
A54:  LIN b9,c9,q by AFF_1:60;
      consider M such that
A55:  q in M and
A56:  K // M by A36,AFF_1:49;
A57:  M is being_line by A56,AFF_1:36;
      not a,b // M
      proof
        assume a,b // M;
        then a,b // K by A56,AFF_1:43;
        then b in K by A36,A34,AFF_1:23;
        hence contradiction by A36,A34,A37,A51,AFF_1:21;
      end;
      then consider p such that
A58:  p in M and
A59:  LIN a,b,p by A57,AFF_1:59;
A60:  a9 in N by AFF_1:15;
A61:  p<>q
      proof
A62:    LIN p,b,a & LIN p,b,b by A59,AFF_1:6,7;
        assume
A63:    p=q;
        then LIN p,b,c by A53,AFF_1:6;
        then p=b by A51,A62,AFF_1:8;
        then LIN b,b9,c9 by A54,A63,AFF_1:6;
        then c9 in P by A10,A11,A15,A50,AFF_1:25;
        hence contradiction by A3,A4,A13,A15,A16,A45,A49,AFF_1:18;
      end;
A64:  c,a // q,p by A34,A37,A55,A56,A58,AFF_1:39;
A65:  LIN b,a,p by A59,AFF_1:6;
A66:  b9<>c9 by A3,A4,A11,A13,A15,A16,A45,A48,AFF_1:18;
A67:  a9<>c9 by A2,A4,A9,A13,A14,A16,A18,A47,AFF_1:18;
      then
A68:  N is being_line by AFF_1:def 3;
A69:  K // N by A20,A35,A67,AFF_1:37;
      then
A70:  N // M by A56,AFF_1:44;
A71:  a9<>b9 by A2,A3,A9,A11,A14,A15,A17,A47,AFF_1:18;
A72:  not LIN a9,b9,c9
      proof
        assume LIN a9,b9,c9;
        then LIN a9,c9,b9 by AFF_1:6;
        then b9 in N by AFF_1:def 2;
        then a9,b9 // N by A68,A60,AFF_1:23;
        then
A73:    a9,b9 // K by A69,AFF_1:43;
        a9,b9 // a,b by A19,AFF_1:4;
        then a,b // K by A71,A73,AFF_1:32;
        then b in K by A36,A34,AFF_1:23;
        hence contradiction by A36,A34,A37,A51,AFF_1:21;
      end;
      not b9,p // N
      proof
        assume b9,p // N;
        then b9,p // M by A70,AFF_1:43;
        then p,b9 // M by AFF_1:34;
        then
A74:    b9 in M by A57,A58,AFF_1:23;
A75:    now
          assume
A76:      b9<>q;
          LIN b9,q,c9 by A54,AFF_1:6;
          then c9 in M by A55,A57,A74,A76,AFF_1:25;
          then a9 in N & b9 in N by A52,A70,A74,AFF_1:15,45;
          hence contradiction by A68,A52,A72,AFF_1:21;
        end;
        now
          assume b9=q;
          then LIN b,b9,c by A53,AFF_1:6;
          then c in P by A10,A11,A15,A50,AFF_1:25;
          hence contradiction by A3,A4,A7,A12,A15,A16,A45,AFF_1:18;
        end;
        hence thesis by A75;
      end;
      then consider x such that
A77:  x in N and
A78:  LIN b9,p,x by A68,AFF_1:59;
      set A9=Line(x,a);
A79:  a<>a9
      proof
        assume
A80:    a=a9;
        ( not LIN o,a,b)& LIN o,b,b9 by A3,A10,A11,A15,A21,AFF_1:6,21;
        hence contradiction by A19,A50,A80,AFF_1:14;
      end;
A81:  x<>a
      proof
        assume x=a;
        then a9 in K by A34,A60,A69,A77,AFF_1:45;
        then A=K by A8,A9,A14,A36,A34,A79,AFF_1:18;
        hence contradiction by A2,A4,A7,A12,A14,A16,A18,A37,AFF_1:18;
      end;
      then
A82:  A9 is being_line by AFF_1:def 3;
A83:  c <>c9
      proof
        assume c =c9;
        then
A84:    c,a // c,a9 by A20,AFF_1:4;
        ( not LIN o,c,a)& LIN o,a,a9 by A2,A8,A9,A14,A21,AFF_1:6,21;
        hence contradiction by A79,A84,AFF_1:14;
      end;
A85:  not LIN b9,c9,x
      proof
A86:    now
A87:      now
            assume q=c9;
            then LIN c,c9,b by A53,AFF_1:6;
            then b in C by A12,A13,A16,A83,AFF_1:25;
            hence contradiction by A3,A4,A6,A10,A15,A16,A45,AFF_1:18;
          end;
          assume c9=x;
          then
A88:      LIN b9,c9,p by A78,AFF_1:6;
          LIN b9,c9,c9 by AFF_1:7;
          then c9 in M by A66,A54,A55,A57,A58,A61,A88,AFF_1:8,25;
          then
A89:      q in N by A55,A52,A70,AFF_1:45;
          LIN q,c9,b9 by A54,AFF_1:6;
          then q=c9 or b9 in N by A68,A52,A89,AFF_1:25;
          hence LIN b9,c9,a9 by A68,A60,A52,A87,AFF_1:21;
        end;
        assume LIN b9,c9,x;
        then
A90:    LIN c9,x,b9 by AFF_1:6;
A91:    LIN c9,x,a9 & LIN c9,x,c9 by A68,A60,A52,A77,AFF_1:21;
        then LIN c9,a9,b9 by A90,A86,AFF_1:6,8;
        then c9,a9 // c9,b9 by AFF_1:def 1;
        then a9,c9 // b9,c9 by AFF_1:4;
        then
A92:    a,c // b9,c9 by A20,A67,AFF_1:5;
        c9=x or LIN b9,c9,a9 by A90,A91,AFF_1:8;
        then b9,c9 // b9,a9 by A86,AFF_1:def 1;
        then b9,c9 // a9,b9 by AFF_1:4;
        then b9,c9 // a,b by A19,A71,AFF_1:5;
        then a,c // a,b by A66,A92,AFF_1:5;
        then LIN a,c,b by AFF_1:def 1;
        hence contradiction by A51,AFF_1:6;
      end;
A93:  x in A9 & a in A9 by AFF_1:15;
      A<>K by A2,A4,A7,A12,A14,A16,A18,A37,AFF_1:18;
      then
A94:  A <> N by A8,A34,A69,AFF_1:45;
A95:  not LIN b,c,a by A51,AFF_1:6;
A96:  p in D by A59,AFF_1:def 2;
A97:  D is being_line & b in D by A38,AFF_1:15,def 3;
A98:  LIN b9,x,p by A78,AFF_1:6;
      c,a // c9,x by A34,A37,A52,A69,A77,AFF_1:39;
      then o in A9 by A1,A3,A4,A6,A7,A10,A11,A12,A13,A15,A16,A45,A53,A54,A98
,A81,A82,A93,A61,A85,A64,A95,A65;
      then x in A by A2,A5,A8,A14,A82,A93,AFF_1:18;
      then x=a9 by A9,A14,A68,A60,A77,A94,AFF_1:18;
      then
A99:  a9 in T & p in T by A98,AFF_1:15,def 2;
      D // T by A19,A38,A71,AFF_1:37;
      then a in D & a9 in D by A96,A99,AFF_1:15,45;
      then b in A by A8,A9,A14,A79,A97,AFF_1:18;
      hence contradiction by A2,A3,A6,A10,A14,A15,A17,AFF_1:18;
    end;
    b9=o implies thesis
    proof
      assume
A100: b9=o;
      LIN o,a,a9 & b,a // b9,a9 by A2,A8,A9,A14,A19,AFF_1:4,21;
      hence thesis by A21,A28,A100,AFF_1:55;
    end;
    hence thesis by A28,A31,A39,A24,A46;
  end;
  hence thesis by A10,A11,A12,A13,A15,AFF_1:51;
end;
