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 satisfying_DES1_3
proof
  assume
A1: AP is satisfying_DES1_2;
  let A,P,C,o,a,a9,b,b9,c,c9,p,q;
  assume that
A2: A is being_line and
A3: P is being_line and
A4: C is being_line and
A5: P<>A and
A6: P<>C and
A7: A<>C and
A8: o in A and
A9: a in A and
A10: a9 in A and
A11: b in P and
A12: b9 in P and
A13: o in C and
A14: c in C and
A15: c9 in C and
A16: o<>a and
A17: o<>b and
A18: o<>c and
A19: p<>q and
A20: not LIN b,a,c and
A21: not LIN b9,a9,c9 and
A22: b<>b9 and
A23: a<>a9 and
A24: LIN b,a,p and
A25: LIN b9,a9,p and
A26: LIN b,c,q and
A27: LIN b9,c9,q and
A28: a,c // a9,c9 and
A29: a,c // p,q;
  set D=Line(b,c), K=Line(b9,a9);
  assume
A30: not thesis;
A31: not LIN o,c,a
  proof
    assume LIN o,c,a;
    then a in C by A4,A13,A14,A18,AFF_1:25;
    hence contradiction by A2,A4,A7,A8,A9,A13,A16,AFF_1:18;
  end;
A32: c <>c9
  proof
    assume c =c9;
    then
A33: c,a // c,a9 by A28,AFF_1:4;
    LIN o,a,a9 by A2,A8,A9,A10,AFF_1:21;
    hence contradiction by A23,A31,A33,AFF_1:14;
  end;
  a<>c by A20,AFF_1:7;
  then
A34: a9,c9 // p,q by A28,A29,AFF_1:5;
A35: p<>b & p<>b9 & q<>b & q<>b9
  proof
A36: now
      assume
A37:  b9=q;
      ( not LIN b9,c9,a9)& c9,a9 // q,p by A21,A34,AFF_1:4,6;
      hence contradiction by A19,A25,A37,AFF_1:55;
    end;
A38: now
      assume
A39:  b=q;
      ( not LIN b,c,a)& c,a // q,p by A20,A29,AFF_1:4,6;
      hence contradiction by A19,A24,A39,AFF_1:55;
    end;
    assume not thesis;
    hence contradiction by A20,A21,A26,A27,A29,A34,A38,A36,AFF_1:55;
  end;
A40: b<>c by A20,AFF_1:7;
  then
A41: D is being_line & c in D by AFF_1:24;
A42: b in D by A40,AFF_1:24;
  then
A43: q in D by A26,A40,A41,AFF_1:25;
A44: now
    assume not C // P;
    then consider x such that
A45: x in C and
A46: x in P by A3,A4,AFF_1:58;
A47: x<>c
    proof
A48:  LIN q,b9,c9 & LIN q,b9,b9 by A27,AFF_1:6,7;
      assume
A49:  x=c;
      then LIN b,c,b9 & LIN b,c,c by A3,A11,A12,A46,AFF_1:21;
      then LIN q,b9,c by A26,A40,AFF_1:8;
      then
A50:  b9 in C by A4,A14,A15,A32,A35,A48,AFF_1:8,25;
      then LIN c,c9,q by A3,A4,A6,A12,A14,A27,A46,A49,AFF_1:18;
      then
A51:  q in C by A4,A14,A15,A32,AFF_1:25;
      c =b9 by A3,A4,A6,A12,A14,A46,A49,A50,AFF_1:18;
      then C=D by A4,A14,A35,A41,A43,A51,AFF_1:18;
      hence contradiction by A3,A4,A6,A11,A12,A22,A42,A50,AFF_1:18;
    end;
A52: x<>b
    proof
A53:  LIN q,c9,b9 by A27,AFF_1:6;
      assume
A54:  x=b;
      then q in C by A4,A14,A26,A40,A45,AFF_1:25;
      then q=c9 or b9 in C by A4,A15,A53,AFF_1:25;
      then c9,a9 // c9,p by A3,A4,A6,A11,A12,A22,A34,A45,A54,AFF_1:4,18;
      then LIN c9,a9,p by AFF_1:def 1;
      then
A55:  LIN p,a9,c9 by AFF_1:6;
      LIN p,a9,b9 & LIN p,a9,a9 by A25,AFF_1:6,7;
      then p=a9 by A21,A55,AFF_1:8;
      then LIN a,a9,b by A24,AFF_1:6;
      then b in A by A2,A9,A10,A23,AFF_1:25;
      hence contradiction by A2,A4,A7,A8,A13,A17,A45,A54,AFF_1:18;
    end;
A56: c,a // q,p & c,a // c9,a9 by A28,A29,AFF_1:4;
    ( not LIN b,c,a)& not LIN b9,c9,a9 by A20,A21,AFF_1:6;
    then x in A by A1,A2,A3,A4,A6,A9,A10,A11,A12,A14,A15,A19,A23,A24,A25,A26
,A27,A45,A46,A47,A52,A56;
    hence contradiction by A2,A4,A7,A8,A13,A30,A45,A46,AFF_1:18;
  end;
A57: a<>b by A20,AFF_1:7;
A58: a9<>b9 by A21,AFF_1:7;
  then
A59: a9 in K by AFF_1:24;
A60: K is being_line & b9 in K by A58,AFF_1:24;
  then
A61: p in K by A25,A58,A59,AFF_1:25;
A62: now
    assume not P // A;
    then consider x such that
A63: x in P and
A64: x in A by A2,A3,AFF_1:58;
A65: x<>b
    proof
      assume
A66:  x=b;
      then p in A by A2,A9,A24,A57,A64,AFF_1:25;
      then a9,c9 // a9,q or b9 in A by A2,A10,A34,A60,A59,A61,AFF_1:18;
      then LIN a9,c9,q by A2,A3,A5,A11,A12,A22,A64,A66,AFF_1:18,def 1;
      then
A67:  LIN q,c9,a9 by AFF_1:6;
      LIN q,c9,b9 & LIN q,c9,c9 by A27,AFF_1:6,7;
      then q=c9 by A21,A67,AFF_1:8;
      then LIN c,c9,b by A26,AFF_1:6;
      then b in C by A4,A14,A15,A32,AFF_1:25;
      hence contradiction by A2,A4,A7,A8,A13,A17,A64,A66,AFF_1:18;
    end;
    x<>a
    proof
      assume x=a;
      then p in P & K<>P by A2,A3,A5,A9,A10,A11,A23,A24,A57,A59,A63,AFF_1:18,25
;
      hence contradiction by A3,A12,A35,A60,A61,AFF_1:18;
    end;
    then x in C by A1,A2,A3,A4,A5,A9,A10,A11,A12,A14,A15,A19,A20,A21,A24,A25
,A26,A27,A28,A29,A32,A63,A64,A65;
    hence contradiction by A2,A4,A7,A8,A13,A30,A63,A64,AFF_1:18;
  end;
  not P // A or not C // P
  proof
    assume not thesis;
    then C // A by AFF_1:44;
    hence contradiction by A7,A8,A13,AFF_1:45;
  end;
  hence contradiction by A62,A44;
end;
