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_DES2_1 iff AP is satisfying_DES2_3
proof
A1: AP is satisfying_DES2_1 implies AP is satisfying_DES2_3
  proof
    assume
A2: AP is satisfying_DES2_1;
    thus AP is satisfying_DES2_3
    proof
      let A,P,C,a,a9,b,b9,c,c9,p,q;
      assume that
A3:   A is being_line and
A4:   P is being_line and
A5:   C is being_line and
A6:   A<>P and
A7:   A<>C and
A8:   P<>C and
A9:   a in A and
A10:  a9 in A and
A11:  b in P and
A12:  b9 in P and
A13:  c in C and
A14:  c9 in C and
A15:  A // P and
A16:  not LIN b,a,c and
A17:  not LIN b9,a9,c9 and
A18:  p<>q and
A19:  c <>c9 and
A20:  LIN b,a,p and
A21:  LIN b9,a9,p and
A22:  LIN b,c,q and
A23:  LIN b9,c9,q and
A24:  a,c // a9,c9 and
A25:  a,c // p,q;
      now
        set A9=Line(a,c), P9=Line(p,q), C9=Line(a9,c9);
A26:    LIN p,a9,b9 by A21,AFF_1:6;
A27:    a<>c by A16,AFF_1:7;
        then
A28:    A9 is being_line & a in A9 by AFF_1:24;
A29:    q<>c
        proof
          assume
A30:      q=c;
          then c,p // c,a by A25,AFF_1:4;
          then LIN c,p,a by AFF_1:def 1;
          then
A31:      LIN p,a,c by AFF_1:6;
          LIN p,a,b & LIN p,a,a by A20,AFF_1:6,7;
          then a=p by A16,A31,AFF_1:8;
          then LIN a,a9,b9 by A21,AFF_1:6;
          then b9 in A or a=a9 by A3,A9,A10,AFF_1:25;
          then a9,c9 // a9,c by A6,A12,A15,A24,AFF_1:4,45;
          then LIN a9,c9,c by AFF_1:def 1;
          then LIN c,c9,a9 by AFF_1:6;
          then
A32:      a9 in C by A5,A13,A14,A19,AFF_1:25;
          LIN c,c9,b9 by A23,A30,AFF_1:6;
          then b9 in C by A5,A13,A14,A19,AFF_1:25;
          hence contradiction by A5,A14,A17,A32,AFF_1:21;
        end;
A33:    a<>p
        proof
          assume a=p;
          then LIN a,c,q by A25,AFF_1:def 1;
          then
A34:      LIN c,q,a by AFF_1:6;
          LIN c,q,b & LIN c,q,c by A22,AFF_1:6,7;
          hence contradiction by A16,A29,A34,AFF_1:8;
        end;
A35:    a<>a9
        proof
A36:      LIN p,a,b & LIN p,a,a by A20,AFF_1:6,7;
          assume
A37:      a=a9;
          then LIN a,c,c9 by A24,AFF_1:def 1;
          then LIN c,c9,a by AFF_1:6;
          then
A38:      a in C by A5,A13,A14,A19,AFF_1:25;
          LIN p,a,b9 by A21,A37,AFF_1:6;
          then b=b9 or a in P by A4,A11,A12,A33,A36,AFF_1:8,25;
          then
A39:      LIN q,b,c9 by A6,A9,A15,A23,AFF_1:6,45;
          LIN q,b,c & LIN q,b,b by A22,AFF_1:6,7;
          then
A40:      q=b or LIN c,c9,b by A39,AFF_1:8;
          not b in C by A5,A13,A16,A38,AFF_1:21;
          then LIN p,q,a by A5,A13,A14,A19,A20,A40,AFF_1:6,25;
          then p,q // p,a by AFF_1:def 1;
          then a,c // p,a by A18,A25,AFF_1:5;
          then a,c // a,p by AFF_1:4;
          then LIN a,c,p by AFF_1:def 1;
          then
A41:      p in C by A5,A13,A27,A38,AFF_1:25;
          LIN p,a,b by A20,AFF_1:6;
          then b in C by A5,A33,A38,A41,AFF_1:25;
          hence contradiction by A5,A13,A16,A38,AFF_1:21;
        end;
A42:    b<>b9
        proof
A43:      p,q // c,a by A25,AFF_1:4;
A44:      LIN q,b,c & LIN q,b,b by A22,AFF_1:6,7;
A45:      LIN p,b,a & LIN p,b,b by A20,AFF_1:6,7;
          assume
A46:      b=b9;
          then LIN p,b,a9 by A21,AFF_1:6;
          then
A47:      p=b or b in A by A3,A9,A10,A35,A45,AFF_1:8,25;
          LIN q,b,c9 by A23,A46,AFF_1:6;
          then b=q or LIN c,c9,b by A44,AFF_1:8;
          then
A48:      b in C by A5,A6,A11,A13,A14,A15,A18,A19,A47,AFF_1:25,45;
          then q in C by A5,A6,A11,A13,A15,A16,A20,A22,A47,AFF_1:25,45;
          then a in C by A5,A6,A11,A13,A15,A18,A47,A48,A43,AFF_1:45,48;
          hence contradiction by A5,A13,A16,A48,AFF_1:21;
        end;
        then
A49:    a9,a // b9,b by A3,A4,A9,A10,A11,A12,A15,A35,AFF_1:38;
A50:    a9<>c9 by A17,AFF_1:7;
        then
A51:    C9 is being_line by AFF_1:24;
A52:    c9 in C9 by A50,AFF_1:24;
A53:    LIN p,a,b by A20,AFF_1:6;
A54:    not LIN p,a9,a
        proof
          assume LIN p,a9,a;
          then
A55:      LIN p,a,a9 by AFF_1:6;
          LIN p,a,a by AFF_1:7;
          then b in A by A3,A9,A10,A33,A35,A53,A55,AFF_1:8,25;
          hence contradiction by A6,A11,A15,AFF_1:45;
        end;
A56:    LIN q,c9,b9 by A23,AFF_1:6;
A57:    a9 in C9 by A50,AFF_1:24;
A58:    c in A9 by A27,AFF_1:24;
        then
A59:    C9 // A9 by A24,A27,A50,A51,A28,A57,A52,AFF_1:38;
A60:    A9<>C9
        proof
          assume
A61:      A9=C9;
          then LIN a,a9,c9 by A28,A57,A52,AFF_1:21;
          then
A62:      c9 in A by A3,A9,A10,A35,AFF_1:25;
          LIN a,a9,c by A28,A58,A57,A61,AFF_1:21;
          then c in A by A3,A9,A10,A35,AFF_1:25;
          hence contradiction by A3,A5,A7,A13,A14,A19,A62,AFF_1:18;
        end;
A63:    LIN q,c,b by A22,AFF_1:6;
A64:    p in P9 by A18,AFF_1:24;
A65:    A9<>P9
        proof
          assume A9=P9;
          then
A66:      LIN p,a,c & LIN p,a,a by A28,A58,A64,AFF_1:21;
          LIN p,a,b by A20,AFF_1:6;
          hence contradiction by A16,A33,A66,AFF_1:8;
        end;
A67:    P9 is being_line by A18,AFF_1:24;
A68:    P9<>C9
        proof
          assume P9=C9;
          then
A69:      LIN p,a9,c9 & LIN p,a9,a9 by A67,A64,A57,A52,AFF_1:21;
          LIN p,a9,b9 by A21,AFF_1:6;
          then p=a9 by A17,A69,AFF_1:8;
          then LIN a,a9,b by A20,AFF_1:6;
          then b in A by A3,A9,A10,A35,AFF_1:25;
          hence contradiction by A6,A11,A15,AFF_1:45;
        end;
A70:    a9,c9 // p,q by A24,A25,A27,AFF_1:5;
A71:    not LIN q,c9,c
        proof
A72:      now
            assume q=c9;
            then c9,a9 // c9,p by A70,AFF_1:4;
            then LIN c9,a9,p by AFF_1:def 1;
            then p in C9 by A50,A51,A57,A52,AFF_1:25;
            then p=a9 or b9 in C9 by A51,A57,A26,AFF_1:25;
            then LIN a,a9,b by A17,A20,A51,A57,A52,AFF_1:6,21;
            then b in A by A3,A9,A10,A35,AFF_1:25;
            hence contradiction by A6,A11,A15,AFF_1:45;
          end;
          assume
A73:      LIN q,c9,c;
          LIN q,c9,c9 by AFF_1:7;
          then
A74:      b9 in C by A5,A13,A14,A19,A56,A73,A72,AFF_1:8,25;
A75:      LIN q,c,c by AFF_1:7;
          LIN q,c,c9 by A73,AFF_1:6;
          then b in C by A5,A13,A14,A19,A29,A63,A75,AFF_1:8,25;
          hence contradiction by A4,A5,A8,A11,A12,A42,A74,AFF_1:18;
        end;
A76:    q in P9 by A18,AFF_1:24;
        then C9 // P9 by A18,A50,A67,A51,A64,A57,A52,A70,AFF_1:38;
        then a9,a // c9,c by A2,A67,A51,A28,A58,A64,A76,A57,A52,A42,A65,A60,A68
,A59,A49,A53,A26,A63,A56,A54,A71;
        hence thesis by A3,A5,A9,A10,A13,A14,A19,A35,AFF_1:38;
      end;
      hence thesis;
    end;
  end;
  AP is satisfying_DES2_3 implies AP is satisfying_DES2_1
  proof
    assume
A77: AP is satisfying_DES2_3;
    thus AP is satisfying_DES2_1
    proof
      let A,P,C,a,a9,b,b9,c,c9,p,q;
      assume that
A78:  A is being_line and
A79:  P is being_line and
A80:  C is being_line and
A81:  A<>P and
      A<>C and
A82:  P<>C and
A83:  a in A and
A84:  a9 in A and
A85:  b in P and
A86:  b9 in P and
A87:  c in C and
A88:  c9 in C and
A89:  A // P and
A90:  A // C and
A91:  not LIN b,a,c and
A92:  not LIN b9,a9,c9 and
A93:  p<>q and
A94:  LIN b,a,p and
A95:  LIN b9,a9,p and
A96:  LIN b,c,q and
A97:  LIN b9,c9,q and
A98:  a,c // p,q;
A99:  C // P by A89,A90,AFF_1:44;
      then
A100: c,c9 // b,b9 by A85,A86,A87,A88,AFF_1:39;
      assume
A101: not thesis;
A102: q<>c
      proof
        assume
A103:   q=c;
        then c,p // c,a by A98,AFF_1:4;
        then LIN c,p,a by AFF_1:def 1;
        then
A104:   LIN p,a,c by AFF_1:6;
        LIN p,a,b & LIN p,a,a by A94,AFF_1:6,7;
        then a=p by A91,A104,AFF_1:8;
        then LIN a,a9,b9 by A95,AFF_1:6;
        then
A105:   b9 in A or a=a9 by A78,A83,A84,AFF_1:25;
        LIN c,c9,b9 by A97,A103,AFF_1:6;
        then c =c9 or b9 in C by A80,A87,A88,AFF_1:25;
        hence contradiction by A81,A82,A86,A89,A101,A99,A105,AFF_1:2,45;
      end;
A106: a<>p
      proof
        assume a=p;
        then LIN a,c,q by A98,AFF_1:def 1;
        then
A107:   LIN c,q,a by AFF_1:6;
        LIN c,q,b & LIN c,q,c by A96,AFF_1:6,7;
        hence contradiction by A91,A102,A107,AFF_1:8;
      end;
A108: a<>a9
      proof
A109:   LIN p,a,b & LIN p,a,a by A94,AFF_1:6,7;
        assume
A110:   a=a9;
        then LIN p,a,b9 by A95,AFF_1:6;
        then a in P or b=b9 by A79,A85,A86,A106,A109,AFF_1:8,25;
        then
A111:   LIN b,q,c9 by A81,A83,A89,A97,AFF_1:6,45;
        LIN b,q,c & LIN b,q,b by A96,AFF_1:6,7;
        then b=q or c =c9 or b in C by A80,A87,A88,A111,AFF_1:8,25;
        then LIN p,q,a by A82,A85,A94,A101,A99,A110,AFF_1:2,6,45;
        then p,q // p,a by AFF_1:def 1;
        then a,c // p,a by A93,A98,AFF_1:5;
        then a,c // a,p by AFF_1:4;
        then LIN a,c,p by AFF_1:def 1;
        then
A112:   LIN p,a,c by AFF_1:6;
        LIN p,a,b & LIN p,a,a by A94,AFF_1:6,7;
        hence contradiction by A91,A106,A112,AFF_1:8;
      end;
A113: not LIN p,a,a9
      proof
        assume
A114:   LIN p,a,a9;
        LIN p,a,b & LIN p,a,a by A94,AFF_1:6,7;
        then b in A by A78,A83,A84,A106,A108,A114,AFF_1:8,25;
        hence contradiction by A81,A85,A89,AFF_1:45;
      end;
      set A9=Line(a,c), P9=Line(p,q), C9=Line(a9,c9);
A115: a<>c by A91,AFF_1:7;
      then
A116: A9 is being_line by AFF_1:24;
A117: c,c9 // a,a9 & LIN q,c,b by A83,A84,A87,A88,A90,A96,AFF_1:6,39;
A118: LIN p,a9,b9 by A95,AFF_1:6;
A119: a in A9 & c in A9 by A115,AFF_1:24;
A120: b<>b9
      proof
        assume b=b9;
        then
A121:   LIN b,p,a9 by A95,AFF_1:6;
        LIN b,p,a & LIN b,p,b by A94,AFF_1:6,7;
        then
A122:   b=p or b in A by A78,A83,A84,A108,A121,AFF_1:8,25;
        then LIN p,q,c by A81,A85,A89,A96,AFF_1:6,45;
        then p,q // p,c by AFF_1:def 1;
        then a,c // p,c by A93,A98,AFF_1:5;
        then c,a // c,p by AFF_1:4;
        then LIN c,a,p by AFF_1:def 1;
        hence contradiction by A81,A85,A89,A91,A122,AFF_1:6,45;
      end;
A123: not LIN q,c,c9
      proof
        assume
A124:   LIN q,c,c9;
        LIN q,c,b & LIN q,c,c by A96,AFF_1:6,7;
        then c =c9 or b in C by A80,A87,A88,A102,A124,AFF_1:8,25;
        then
A125:   LIN q,c,b9 by A82,A85,A97,A99,AFF_1:6,45;
        LIN q,c,b & LIN q,c,c by A96,AFF_1:6,7;
        then c in P by A79,A85,A86,A102,A120,A125,AFF_1:8,25;
        hence contradiction by A82,A87,A99,AFF_1:45;
      end;
A126: LIN q,c9,b9 & LIN p,a,b by A94,A97,AFF_1:6;
A127: a9<>c9 by A92,AFF_1:7;
      then
A128: C9 is being_line by AFF_1:24;
A129: p in P9 by A93,AFF_1:24;
A130: A9<>P9
      proof
        assume A9=P9;
        then
A131:   LIN p,a,c & LIN p,a,a by A116,A119,A129,AFF_1:21;
        LIN p,a,b by A94,AFF_1:6;
        hence contradiction by A91,A106,A131,AFF_1:8;
      end;
A132: P9 is being_line & q in P9 by A93,AFF_1:24;
      then
A133: A9 // P9 by A93,A98,A115,A116,A119,A129,AFF_1:38;
A134: a9 in C9 & c9 in C9 by A127,AFF_1:24;
      then A9<>C9 by A101,A116,A119,AFF_1:51;
      then A9 // C9 by A77,A127,A116,A128,A119,A129,A132,A134,A100,A133,A130
,A120,A123,A113,A117,A126,A118;
      hence contradiction by A101,A119,A134,AFF_1:39;
    end;
  end;
  hence thesis by A1;
end;
