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_3 implies AP is satisfying_DES2_1
proof
  assume
A1: AP is satisfying_DES1_3;
  let A,P,C,a,a9,b,b9,c,c9,p,q such that
A2: A is being_line and
A3: P is being_line and
A4: C is being_line and
A5: A<>P and
A6: A<>C and
A7: P<>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 // P and
A15: A // C and
A16: not LIN b,a,c and
A17: not LIN b9,a9,c9 and
A18: p<>q and
A19: LIN b,a,p and
A20: LIN b9,a9,p and
A21: LIN b,c,q and
A22: LIN b9,c9,q and
A23: a,c // p,q;
A24: P // C by A14,A15,AFF_1:44;
  set K=Line(p,q), M=Line(a,c), N=Line(a9,c9);
A25: a<>c by A16,AFF_1:7;
  then
A26: a in M by AFF_1:24;
A27: c,c9 // a,a9 & LIN q,c,b by A8,A9,A12,A13,A15,A21,AFF_1:6,39;
A28: LIN p,a9,b9 by A20,AFF_1:6;
  C // P by A14,A15,AFF_1:44;
  then
A29: c,c9 // b,b9 by A10,A11,A12,A13,AFF_1:39;
A30: LIN q,c9,b9 & LIN p,a,b by A19,A22,AFF_1:6;
A31: c in M by A25,AFF_1:24;
  assume
A32: not thesis;
A33: c <>q
  proof
    assume
A34: c =q;
    then c,a // c,p by A23,AFF_1:4;
    then LIN c,a,p by AFF_1:def 1;
    then
A35: LIN p,a,c by AFF_1:6;
    LIN p,a,b & LIN p,a,a by A19,AFF_1:6,7;
    then p=a by A16,A35,AFF_1:8;
    then LIN a,a9,b9 by A20,AFF_1:6;
    then
A36: a=a9 or b9 in A by A2,A8,A9,AFF_1:25;
    LIN c,c9,b9 by A22,A34,AFF_1:6;
    then c =c9 or b9 in C by A4,A12,A13,AFF_1:25;
    hence contradiction by A5,A7,A11,A14,A32,A24,A36,AFF_1:2,45;
  end;
A37: c <>c9
  proof
A38: now
      assume
A39:  p=b;
      LIN b,q,c by A21,AFF_1:6;
      then b,q // b,c by AFF_1:def 1;
      then a,c // b,c or b=q by A23,A39,AFF_1:5;
      then c,a // c,b by A18,A39,AFF_1:4;
      then LIN c,a,b by AFF_1:def 1;
      hence contradiction by A16,AFF_1:6;
    end;
A40: LIN q,c,b & LIN q,c,c by A21,AFF_1:6,7;
    assume
A41: c =c9;
    then LIN q,c,b9 by A22,AFF_1:6;
    then b=b9 or c in P by A3,A10,A11,A33,A40,AFF_1:8,25;
    then
A42: LIN p,b,a9 by A7,A12,A20,A24,AFF_1:6,45;
    LIN p,b,a & LIN p,b,b by A19,AFF_1:6,7;
    then a=a9 or b in A by A2,A8,A9,A42,A38,AFF_1:8,25;
    hence contradiction by A5,A10,A14,A32,A41,AFF_1:2,45;
  end;
A43: b<>b9
  proof
    assume b=b9;
    then
A44: LIN q,b,c9 by A22,AFF_1:6;
    LIN q,b,c & LIN q,b,b by A21,AFF_1:6,7;
    then
A45: q=b or b in C by A4,A12,A13,A37,A44,AFF_1:8,25;
    b,a // b,p by A19,AFF_1:def 1;
    then a,b // p,b by AFF_1:4;
    then a,c // a,b or p=b by A7,A10,A23,A24,A45,AFF_1:5,45;
    then LIN a,c,b by A7,A10,A18,A24,A45,AFF_1:45,def 1;
    hence contradiction by A16,AFF_1:6;
  end;
A46: not LIN q,c,c9
  proof
    assume
A47: LIN q,c,c9;
    LIN q,c,b & LIN q,c,c by A21,AFF_1:6,7;
    then b in C by A4,A12,A13,A33,A37,A47,AFF_1:8,25;
    hence contradiction by A7,A10,A24,AFF_1:45;
  end;
A48: a9<>c9 by A17,AFF_1:7;
  then
A49: N is being_line by AFF_1:24;
A50: p<>a
  proof
    assume p=a;
    then LIN a,c,q by A23,AFF_1:def 1;
    then
A51: LIN c,q,a by AFF_1:6;
    LIN c,q,b & LIN c,q,c by A21,AFF_1:6,7;
    hence contradiction by A16,A33,A51,AFF_1:8;
  end;
A52: not LIN p,a,a9
  proof
    assume
A53: LIN p,a,a9;
    LIN p,a,b & LIN p,a,a by A19,AFF_1:6,7;
    then a=a9 or b in A by A2,A8,A9,A50,A53,AFF_1:8,25;
    then
A54: LIN p,a,b9 by A5,A10,A14,A20,AFF_1:6,45;
    LIN p,a,b & LIN p,a,a by A19,AFF_1:6,7;
    then a in P by A3,A10,A11,A43,A50,A54,AFF_1:8,25;
    hence contradiction by A5,A8,A14,AFF_1:45;
  end;
A55: M is being_line by A25,AFF_1:24;
A56: K is being_line & q in K by A18,AFF_1:24;
A57: now
    assume M=K;
    then
A58: LIN q,c,a & LIN q,c,c by A56,A26,A31,AFF_1:21;
    LIN q,c,b by A21,AFF_1:6;
    hence contradiction by A16,A33,A58,AFF_1:8;
  end;
A59: c9 in N by A48,AFF_1:24;
A60: a9 in N by A48,AFF_1:24;
  then consider x such that
A61: x in M and
A62: x in N by A32,A55,A49,A26,A31,A59,AFF_1:39,58;
A63: now
    assume x=c;
    then N=C by A4,A12,A13,A37,A49,A59,A62,AFF_1:18;
    hence contradiction by A6,A9,A15,A60,AFF_1:45;
  end;
A64: p in K by A18,AFF_1:24;
  then
A65: M // K by A18,A23,A25,A55,A56,A26,A31,AFF_1:38;
A66: now
    assume x=c9;
    then M=C by A4,A12,A13,A37,A55,A31,A61,AFF_1:18;
    hence contradiction by A6,A8,A15,A26,AFF_1:45;
  end;
  M<>N by A32,A55,A26,A31,A60,A59,AFF_1:51;
  then x in K by A1,A18,A25,A55,A49,A64,A56,A26,A31,A60,A59,A61,A62,A29,A27,A30
,A28,A43,A63,A66,A46,A52;
  hence contradiction by A65,A61,A57,AFF_1:45;
end;
