reserve X for OrtAfPl;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,c3,d,d1,d2,d3,d4,e1,e2 for
  Element of X;
reserve a29,a39,b29,x9 for Element of the AffinStruct of X;
reserve A,K,M,N for Subset of X;
reserve A9,K9 for Subset of the AffinStruct of X;

theorem
  X is satisfying_MH1 & X is satisfying_MH2 implies X is satisfying_OPAP
proof
  assume
A1: X is satisfying_MH1;
  assume
A2: X is satisfying_MH2;
  let o,a1,a2,a3,b1,b2,b3,M,N;
  assume that
A3: o in M and
A4: a1 in M and
A5: a2 in M and
A6: a3 in M and
A7: o in N and
A8: b1 in N and
A9: b2 in N and
A10: b3 in N and
A11: not b2 in M and
A12: not a3 in N and
A13: M _|_ N and
A14: o<>a1 and
A15: o<>a2 and
  o<>a3 and
A16: o<>b1 and
  o<>b2 and
A17: o<>b3 and
A18: a3,b2 // a2,b1 and
A19: a3,b3 // a1,b1;
  reconsider o9=o,a19=a1,a29=a2,a39=a3,b19=b1,b29=b2,b39=b3
   as Element of the AffinStruct of X;
  reconsider M9=M,N9=N as Subset of the AffinStruct of X;
  N is being_line by A13,ANALMETR:44;
  then
A20: N9 is being_line by ANALMETR:43;
  M is being_line by A13,ANALMETR:44;
  then
A21: M9 is being_line by ANALMETR:43;
A22: now
    consider e1 be Element of X such that
A23: o,a3 // o,e1 and
A24: o<>e1 by ANALMETR:39;
    consider d1 such that
A25: o,b3 // o,d1 and
A26: o<>d1 by ANALMETR:39;
    reconsider d19=d1 as Element of the AffinStruct of X;
    consider e2 be Element of X such that
A27: a1,b3 _|_ d1,e2 and
A28: d1<>e2 by ANALMETR:39;
    reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
    assume that
A29: a1<>a2 and
    a2<>a3 and
A30: a1<>a3;
A31: LIN o,b3,d1 by A25,ANALMETR:def 10;
    then
A32: LIN o9,b39,d19 by ANALMETR:40;
    N is being_line by A13,ANALMETR:44;
    then
A33: N9 is being_line by ANALMETR:43;
    then
A34: o9,b29 // o9,b39 by A7,A9,A10,AFF_1:39,41;
    then o,b2 // o,b3 by ANALMETR:36;
    then
A35: LIN o,b2,b3 by ANALMETR:def 10;
A36: b2<>b3
    proof
A37:  not LIN o9,b19,a29
      proof
        o9,a29 // o9,a39 by A3,A5,A6,A21,AFF_1:39,41;
        then
A38:    LIN o9,a29,a39 by AFF_1:def 1;
        assume LIN o9,b19,a29;
        then a29 in N9 by A7,A8,A16,A20,AFF_1:25;
        hence contradiction by A7,A12,A15,A20,A38,AFF_1:25;
      end;
      assume b2=b3;
      then a1,b1 // a2,b1 by A6,A11,A18,A19,ANALMETR:60;
      then b1,a2 // b1,a1 by ANALMETR:59;
      then
A39:  b19,a29 // b19,a19 by ANALMETR:36;
      o9,a29 // o9,a19 by A3,A4,A5,A21,AFF_1:39,41;
      then LIN o9,a29,a19 by AFF_1:def 1;
      hence contradiction by A29,A37,A39,AFF_1:14;
    end;
A40: not LIN b2,a3,b3
    proof
      assume LIN b2,a3,b3;
      then LIN b29,a39,b39 by ANALMETR:40;
      then LIN b29,b39,a39 by AFF_1:6;
      hence contradiction by A9,A10,A12,A36,A33,AFF_1:25;
    end;
A41: a3<>b3
    proof
      assume a3=b3;
      then LIN b29,a39,b39 by AFF_1:7;
      hence contradiction by A40,ANALMETR:40;
    end;
    b29,b39 // b29,b19 by A8,A9,A10,A33,AFF_1:39,41;
    then b2,b3 // b2,b1 by ANALMETR:36;
    then
A42: LIN b2,b3,b1 by ANALMETR:def 10;
    M is being_line by A13,ANALMETR:44;
    then
A43: M9 is being_line by ANALMETR:43;
    then o9,a39 // o9,a19 by A3,A4,A6,AFF_1:39,41;
    then
A44: o,a3 // o,a1 by ANALMETR:36;
    then
A45: LIN o,a3,a1 by ANALMETR:def 10;
A46: not LIN a3,a1,b2
    proof
      assume LIN a3,a1,b2;
      then LIN a39,a19,b29 by ANALMETR:40;
      hence contradiction by A4,A6,A11,A30,A43,AFF_1:25;
    end;
A47: a3<>b2
    proof
      assume a3=b2;
      then LIN a39,a19,b29 by AFF_1:7;
      hence contradiction by A46,ANALMETR:40;
    end;
A48: o,a3 _|_ o,b3 by A3,A6,A7,A10,A13,ANALMETR:56;
    not o,e1 // d1,e2
    proof
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      assume o,e1 // d1,e2;
      then o9,e19 // d19,e29 by ANALMETR:36;
      then d19,e29 // o9,e19 by AFF_1:4;
      then d1,e2 // o,e1 by ANALMETR:36;
      then o,e1 _|_ a1,b3 by A27,A28,ANALMETR:62;
      then o,a3 _|_ a1,b3 by A23,A24,ANALMETR:62;
      then o,b3 // a1,b3 by A7,A12,A48,ANALMETR:63;
      then o9,b39 // a19,b39 by ANALMETR:36;
      then b39,o9 // b39,a19 by AFF_1:4;
      then LIN b39,o9,a19 by AFF_1:def 1;
      then LIN o9,b39,a19 by AFF_1:6;
      then
A49:  o9,b39// o9,a19 by AFF_1:def 1;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then o9,b29 // o9,b39 by AFF_1:def 1;
      then o9,b39 // o9,b29 by AFF_1:4;
      then o9,a19 // o9,b29 by A17,A49,DIRAF:40;
      then LIN o9,a19,b29 by AFF_1:def 1;
      then LIN a19,o9,b29 by AFF_1:6;
      then
A50:  a19,o9 // a19,b29 by AFF_1:def 1;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a19,o9,a39 by AFF_1:6;
      then a19,o9 // a19,a39 by AFF_1:def 1;
      then a19,b29 // a19,a39 by A14,A50,DIRAF:40;
      then LIN a19,b29,a39 by AFF_1:def 1;
      then LIN a39,a19,b29 by AFF_1:6;
      hence contradiction by A46,ANALMETR:40;
    end;
    then not o9,e19 // d19,e29 by ANALMETR:36;
    then consider d29 be Element of the AffinStruct of X such that
A51: LIN o9,e19,d29 and
A52: LIN d19,e29,d29 by AFF_1:60;
    reconsider d2=d29 as Element of X;
    LIN d1,e2,d2 by A52,ANALMETR:40;
    then
A53: d1,e2 // d1,d2 by ANALMETR:def 10;
    then
A54: a1,b3 _|_ d1,d2 by A27,A28,ANALMETR:62;
    then
A55: d1,d2 _|_ b3,a1 by ANALMETR:61;
    LIN o,e1,d2 by A51,ANALMETR:40;
    then o,e1 // o,d2 by ANALMETR:def 10;
    then
A56: o,a3 // o,d2 by A23,A24,ANALMETR:60;
    then
A57: LIN o,a3,d2 by ANALMETR:def 10;
    then
A58: LIN o9,a39,d29 by ANALMETR:40;
    consider e1 be Element of X such that
A59: o,b3 // o,e1 and
A60: o<>e1 by ANALMETR:39;
    consider e2 be Element of X such that
A61: b3,a3 _|_ d2,e2 and
A62: d2<>e2 by ANALMETR:39;
    reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
    not o,e1 // d2,e2
    proof
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      assume o,e1 // d2,e2;
      then o9,e19 // d29,e29 by ANALMETR:36;
      then d29,e29 // o9,e19 by AFF_1:4;
      then d2,e2 // o,e1 by ANALMETR:36;
      then o,e1 _|_ b3,a3 by A61,A62,ANALMETR:62;
      then o,b3 _|_ b3,a3 by A59,A60,ANALMETR:62;
      then b3,a3 // o,a3 by A17,A48,ANALMETR:63;
      then b39,a39 // o9,a39 by ANALMETR:36;
      then a39,b39 // a39,o9 by AFF_1:4;
      then LIN a39,b39,o9 by AFF_1:def 1;
      then LIN b39,o9,a39 by AFF_1:6;
      then
A63:  b39,o9 // b39,a39 by AFF_1:def 1;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b39,o9,b29 by AFF_1:6;
      then b39,o9 // b39,b29 by AFF_1:def 1;
      then b39,a39 // b39,b29 by A17,A63,DIRAF:40;
      then LIN b39,a39,b29 by AFF_1:def 1;
      then LIN b29,a39,b39 by AFF_1:6;
      hence contradiction by A40,ANALMETR:40;
    end;
    then not o9,e19 // d29,e29 by ANALMETR:36;
    then consider d39 be Element of the AffinStruct of X such that
A64: LIN o9,e19,d39 and
A65: LIN d29,e29,d39 by AFF_1:60;
    reconsider d3=d39 as Element of X;
    LIN d2,e2,d3 by A65,ANALMETR:40;
    then
A66: d2,e2 // d2,d3 by ANALMETR:def 10;
    then
A67: b3,a3 _|_ d2,d3 by A61,A62,ANALMETR:62;
    then d2,d3 _|_ a3,b3 by ANALMETR:61;
    then
A68: d2,d3 _|_ a1,b1 by A19,A41,ANALMETR:62;
    LIN o,e1,d3 by A64,ANALMETR:40;
    then o,e1 // o,d3 by ANALMETR:def 10;
    then
A69: o,b3 // o,d3 by A59,A60,ANALMETR:60;
    then
A70: LIN o,b3,d3 by ANALMETR:def 10;
    then
A71: LIN o9,b39,d39 by ANALMETR:40;
    consider e2 be Element of X such that
A72: a3,b2 _|_ d3,e2 and
A73: d3<>e2 by ANALMETR:39;
    consider e1 be Element of X such that
A74: o,a3 // o,e1 and
A75: o<>e1 by ANALMETR:39;
    reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
    not o,e1 // d3,e2
    proof
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b29,o9,b39 by AFF_1:6;
      then
A76:  b29,o9 // b29,b39 by AFF_1:def 1;
      assume o,e1 // d3,e2;
      then o9,e19 // d39,e29 by ANALMETR:36;
      then d39,e29 // o9,e19 by AFF_1:4;
      then d3,e2 // o,e1 by ANALMETR:36;
      then o,e1 _|_ a3,b2 by A72,A73,ANALMETR:62;
      then o,a3 _|_ a3,b2 by A74,A75,ANALMETR:62;
      then
A77:  o,b3 // a3,b2 by A7,A12,A48,ANALMETR:63;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN o9,b39,b29 by AFF_1:6;
      then LIN o,b3,b2 by ANALMETR:40;
      then o,b3 // o,b2 by ANALMETR:def 10;
      then o,b2 // a3,b2 by A17,A77,ANALMETR:60;
      then o9,b29 // a39,b29 by ANALMETR:36;
      then b29,o9 // b29,a39 by AFF_1:4;
      then b29,a39 // b29,b39 by A3,A11,A76,DIRAF:40;
      then LIN b29,a39,b39 by AFF_1:def 1;
      hence contradiction by A40,ANALMETR:40;
    end;
    then not o9,e19 // d39,e29 by ANALMETR:36;
    then consider d49 be Element of the AffinStruct of X such that
A78: LIN o9,e19,d49 and
A79: LIN d39,e29,d49 by AFF_1:60;
    reconsider d4=d49 as Element of X;
    LIN d3,e2,d4 by A79,ANALMETR:40;
    then
A80: d3,e2 // d3,d4 by ANALMETR:def 10;
    then
A81: d3,d4 _|_ a3,b2 by A72,A73,ANALMETR:62;
    LIN o,e1,d4 by A78,ANALMETR:40;
    then o,e1 // o,d4 by ANALMETR:def 10;
    then
A82: o,a3 // o,d4 by A74,A75,ANALMETR:60;
    then
A83: LIN o,a3,d4 by ANALMETR:def 10;
    then
A84: LIN o9,a39,d49 by ANALMETR:40;
A85: a3,b2 _|_ d3,d4 by A72,A73,A80,ANALMETR:62;
    then d3,d4 _|_ a2,b1 or a3=b2 by A18,ANALMETR:62;
    then
A86: d3,d4 _|_ b1,a2 by A47,ANALMETR:61;
A87: d2<>o
    proof
      o,a3 _|_ o,d1 by A17,A48,A25,ANALMETR:62;
      then
A88:  d1,o _|_ o,a3 by ANALMETR:61;
      assume d2=o;
      then d1,o _|_ a1,b3 by A27,A28,A53,ANALMETR:62;
      then o,a3 // a1,b3 by A26,A88,ANALMETR:63;
      then a1,b3 // o,a1 by A7,A12,A44,ANALMETR:60;
      then a19,b39 // o9,a19 by ANALMETR:36;
      then a19,b39 // a19,o9 by AFF_1:4;
      then LIN a19,b39,o9 by AFF_1:def 1;
      then LIN o9,b39,a19 by AFF_1:6;
      then
A89:  o9,b39 // o9,a19 by AFF_1:def 1;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN o9,b39,b29 by AFF_1:6;
      then o9,b39 // o9,b29 by AFF_1:def 1;
      then o9,a19 // o9,b29 by A17,A89,DIRAF:40;
      then LIN o9,a19,b29 by AFF_1:def 1;
      then LIN a19,o9,b29 by AFF_1:6;
      then
A90:  a19,o9 // a19,b29 by AFF_1:def 1;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a19,o9,a39 by AFF_1:6;
      then a19,o9 // a19,a39 by AFF_1:def 1;
      then a19,b29 // a19,a39 by A14,A90,DIRAF:40;
      then LIN a19,b29,a39 by AFF_1:def 1;
      then LIN a39,a19,b29 by AFF_1:6;
      hence contradiction by A46,ANALMETR:40;
    end;
A91: not LIN d1,d2,d3
    proof
A92:  d2<>d3
      proof
        assume d2=d3;
        then o9,b39 // o9,d29 by A69,ANALMETR:36;
        then o9,d29 // o9,b39 by AFF_1:4;
        then o,d2 // o,b3 by ANALMETR:36;
        then o,b3 // o,a3 by A56,A87,ANALMETR:60;
        then o9,b39 // o9,a39 by ANALMETR:36;
        then LIN o9,b39,a39 by AFF_1:def 1;
        then LIN b39,o9,a39 by AFF_1:6;
        then b39,o9 // b39,a39 by AFF_1:def 1;
        then
A93:    b3,o // b3,a3 by ANALMETR:36;
        LIN o9,b29,b39 by A35,ANALMETR:40;
        then LIN b39,o9,b29 by AFF_1:6;
        then b39,o9 // b39,b29 by AFF_1:def 1;
        then b3,o // b3,b2 by ANALMETR:36;
        then b3,b2 // b3,a3 by A17,A93,ANALMETR:60;
        then LIN b3,b2,a3 by ANALMETR:def 10;
        then LIN b39,b29,a39 by ANALMETR:40;
        then LIN b29,a39,b39 by AFF_1:6;
        hence contradiction by A40,ANALMETR:40;
      end;
A94:  d1<>d2
      proof
        assume d1=d2;
        then o9,a39 // o9,d19 by A56,ANALMETR:36;
        then o9,d19 // o9,a39 by AFF_1:4;
        then o,d1 // o,a3 by ANALMETR:36;
        then o,b3 // o,a3 by A25,A26,ANALMETR:60;
        then o9,b39 // o9,a39 by ANALMETR:36;
        then LIN o9,b39,a39 by AFF_1:def 1;
        then LIN b39,o9,a39 by AFF_1:6;
        then b39,o9 // b39,a39 by AFF_1:def 1;
        then
A95:    b3,o // b3,a3 by ANALMETR:36;
        LIN o9,b29,b39 by A35,ANALMETR:40;
        then LIN b39,o9,b29 by AFF_1:6;
        then b39,o9 // b39,b29 by AFF_1:def 1;
        then b3,o // b3,b2 by ANALMETR:36;
        then b3,b2 // b3,a3 by A17,A95,ANALMETR:60;
        then LIN b3,b2,a3 by ANALMETR:def 10;
        then LIN b39,b29,a39 by ANALMETR:40;
        then LIN b29,a39,b39 by AFF_1:6;
        hence contradiction by A40,ANALMETR:40;
      end;
      assume LIN d1,d2,d3;
      then LIN d19,d29,d39 by ANALMETR:40;
      then LIN d29,d19,d39 by AFF_1:6;
      then d29,d19 // d29,d39 by AFF_1:def 1;
      then d19,d29 // d29,d39 by AFF_1:4;
      then d1,d2 // d2,d3 by ANALMETR:36;
      then d2,d3 _|_ a1,b3 by A54,A94,ANALMETR:62;
      then a1,b3 // b3,a3 by A67,A92,ANALMETR:63;
      then a19,b39 // b39,a39 by ANALMETR:36;
      then b39,a19 // b39,a39 by AFF_1:4;
      then LIN b39,a19,a39 by AFF_1:def 1;
      then LIN a19,a39,b39 by AFF_1:6;
      then a19,a39 // a19,b39 by AFF_1:def 1;
      then
A96:  a1,a3 // a1,b3 by ANALMETR:36;
A97:  a1<>a3
      proof
        assume a1=a3;
        then LIN a39,a19,b29 by AFF_1:7;
        hence contradiction by A46,ANALMETR:40;
      end;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a19,a39,o9 by AFF_1:6;
      then a19,a39 // a19,o9 by AFF_1:def 1;
      then a1,a3 // a1,o by ANALMETR:36;
      then a1,o // a1,b3 by A97,A96,ANALMETR:60;
      then LIN a1,o,b3 by ANALMETR:def 10;
      then LIN a19,o9,b39 by ANALMETR:40;
      then LIN o9,b39,a19 by AFF_1:6;
      then
A98:  o9,b39 // o9,a19 by AFF_1:def 1;
      o9,b39 // o9,b29 by A34,AFF_1:4;
      then o9,a19 // o9,b29 by A17,A98,DIRAF:40;
      then LIN o9,a19,b29 by AFF_1:def 1;
      then LIN a19,o9,b29 by AFF_1:6;
      then a19,o9 // a19,b29 by AFF_1:def 1;
      then
A99:  a1,o // a1,b2 by ANALMETR:36;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a19,o9,a39 by AFF_1:6;
      then a19,o9 // a19,a39 by AFF_1:def 1;
      then a1,o // a1,a3 by ANALMETR:36;
      then a1,b2 // a1,a3 by A14,A99,ANALMETR:60;
      then LIN a1,b2,a3 by ANALMETR:def 10;
      then LIN a19,b29,a39 by ANALMETR:40;
      then LIN a39,a19,b29 by AFF_1:6;
      hence contradiction by A46,ANALMETR:40;
    end;
A100: d2<>d4
    proof
A101: d2<>d3
      proof
        assume d2=d3;
        then LIN d19,d29,d39 by AFF_1:7;
        hence contradiction by A91,ANALMETR:40;
      end;
      assume d2=d4;
      then a3,b2 _|_ d2,d3 by A85,ANALMETR:61;
      then a3,b2 // b3,a3 by A67,A101,ANALMETR:63;
      then a3,b2 // a3,b3 by ANALMETR:59;
      then LIN a3,b2,b3 by ANALMETR:def 10;
      then LIN a39,b29,b39 by ANALMETR:40;
      then LIN b29,b39,a39 by AFF_1:6;
      hence contradiction by A9,A10,A12,A20,A36,AFF_1:25;
    end;
    LIN o9,b39,b39 by AFF_1:7;
    then
A102: LIN d19,d39,b39 by A17,A32,A71,AFF_1:8;
    then consider A9 such that
A103: A9 is being_line and
A104: d19 in A9 and
A105: d39 in A9 and
A106: b39 in A9 by AFF_1:21;
    reconsider A=A9 as Subset of X;
A107: d19<>d39
    proof
      assume d19=d39;
      then LIN d19,d29,d39 by AFF_1:7;
      hence contradiction by A91,ANALMETR:40;
    end;
    then A9=Line(d19,d39) by A103,A104,A105,AFF_1:24;
    then
A108: A=Line(d1,d3) by ANALMETR:41;
    LIN o9,b29,b39 by A35,ANALMETR:40;
    then
A109: LIN o9,b39,b29 by AFF_1:6;
    then
A110: b2 in A by A17,A32,A71,A103,A104,A105,A107,AFF_1:8,25;
A111: o<>d3
    proof
      assume d3=o;
      then
A112: d2,o _|_ b3,a3 by A61,A62,A66,ANALMETR:62;
      o,b3 _|_ o,d2 by A7,A12,A48,A56,ANALMETR:62;
      then d2,o _|_ o,b3 by ANALMETR:61;
      then o,b3 // b3,a3 by A87,A112,ANALMETR:63;
      then o9,b39 // b39,a39 by ANALMETR:36;
      then
A113: b39,o9 // b39,a39 by AFF_1:4;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b39,o9,b29 by AFF_1:6;
      then b39,o9 // b39,b29 by AFF_1:def 1;
      then b39,a39 // b39,b29 by A17,A113,DIRAF:40;
      then LIN b39,a39,b29 by AFF_1:def 1;
      then LIN b29,a39,b39 by AFF_1:6;
      hence contradiction by A40,ANALMETR:40;
    end;
A114: o<>d4
    proof
      o,a3 _|_ o,d3 by A17,A48,A69,ANALMETR:62;
      then
A115: d3,o _|_ o,a3 by ANALMETR:61;
      assume d4=o;
      then d3,o _|_ a3,b2 by A72,A73,A80,ANALMETR:62;
      then o,a3 // a3,b2 by A111,A115,ANALMETR:63;
      then o9,a39 // a39,b29 by ANALMETR:36;
      then
A116: a39,o9 // a39,b29 by AFF_1:4;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a39,o9,a19 by AFF_1:6;
      then a39,o9 // a39,a19 by AFF_1:def 1;
      then a39,b29 // a39,a19 by A7,A12,A116,DIRAF:40;
      then LIN a39,b29,a19 by AFF_1:def 1;
      then LIN a39,a19,b29 by AFF_1:6;
      hence contradiction by A46,ANALMETR:40;
    end;
A117: not LIN d1,d4,d3
    proof
A118: d1<>d3
      proof
A119:   d1<>d2
        proof
          assume d1=d2;
          then o9,a39 // o9,d19 by A56,ANALMETR:36;
          then o9,d19 // o9,a39 by AFF_1:4;
          then o,d1 // o,a3 by ANALMETR:36;
          then o,b3 // o,a3 by A25,A26,ANALMETR:60;
          then o9,b39 // o9,a39 by ANALMETR:36;
          then LIN o9,b39,a39 by AFF_1:def 1;
          then LIN b39,o9,a39 by AFF_1:6;
          then b39,o9 // b39,a39 by AFF_1:def 1;
          then
A120:     b3,o // b3,a3 by ANALMETR:36;
          LIN o9,b29,b39 by A35,ANALMETR:40;
          then LIN b39,o9,b29 by AFF_1:6;
          then b39,o9 // b39,b29 by AFF_1:def 1;
          then b3,o // b3,b2 by ANALMETR:36;
          then b3,b2 // b3,a3 by A17,A120,ANALMETR:60;
          then LIN b3,b2,a3 by ANALMETR:def 10;
          then LIN b39,b29,a39 by ANALMETR:40;
          then LIN b29,a39,b39 by AFF_1:6;
          hence contradiction by A40,ANALMETR:40;
        end;
        assume d1=d3;
        then a3,b3 _|_ d1,d2 by A67,ANALMETR:61;
        then a1,b3 // a3,b3 by A54,A119,ANALMETR:63;
        then a19,b39 // a39,b39 by ANALMETR:36;
        then b39,a19 // b39,a39 by AFF_1:4;
        then LIN b39,a19,a39 by AFF_1:def 1;
        then LIN a19,a39,b39 by AFF_1:6;
        then a19,a39 // a19,b39 by AFF_1:def 1;
        then
A121:   a1,a3 // a1,b3 by ANALMETR:36;
A122:   a1<>a3
        proof
          assume a1=a3;
          then LIN a39,a19,b29 by AFF_1:7;
          hence contradiction by A46,ANALMETR:40;
        end;
        LIN o9,a39,a19 by A45,ANALMETR:40;
        then LIN a19,a39,o9 by AFF_1:6;
        then a19,a39 // a19,o9 by AFF_1:def 1;
        then a1,a3 // a1,o by ANALMETR:36;
        then a1,o // a1,b3 by A122,A121,ANALMETR:60;
        then LIN a1,o,b3 by ANALMETR:def 10;
        then LIN a19,o9,b39 by ANALMETR:40;
        then LIN o9,b39,a19 by AFF_1:6;
        then
A123:   o9,b39 // o9,a19 by AFF_1:def 1;
        o9,b39 // o9,b29 by A34,AFF_1:4;
        then o9,a19 // o9,b29 by A17,A123,DIRAF:40;
        then LIN o9,a19,b29 by AFF_1:def 1;
        then LIN a19,o9,b29 by AFF_1:6;
        then a19,o9 // a19,b29 by AFF_1:def 1;
        then
A124:   a1,o // a1,b2 by ANALMETR:36;
        LIN o9,a39,a19 by A45,ANALMETR:40;
        then LIN a19,o9,a39 by AFF_1:6;
        then a19,o9 // a19,a39 by AFF_1:def 1;
        then a1,o // a1,a3 by ANALMETR:36;
        then a1,b2 // a1,a3 by A14,A124,ANALMETR:60;
        then LIN a1,b2,a3 by ANALMETR:def 10;
        then LIN a19,b29,a39 by ANALMETR:40;
        then LIN a39,a19,b29 by AFF_1:6;
        hence contradiction by A46,ANALMETR:40;
      end;
      assume LIN d1,d4,d3;
      then LIN d19,d49,d39 by ANALMETR:40;
      then
A125: LIN d19,d39,d49 by AFF_1:6;
A126: b29<>b39
      proof
        assume b29=b39;
        then LIN b29,a39,b39 by AFF_1:7;
        hence contradiction by A40,ANALMETR:40;
      end;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b29,b39,o9 by AFF_1:6;
      then
A127: b29,b39 // b29,o9 by AFF_1:def 1;
      LIN d19,d39,b29 by A17,A32,A71,A109,AFF_1:8;
      then LIN b29,b39,d49 by A102,A118,A125,AFF_1:8;
      then b29,b39 // b29,d49 by AFF_1:def 1;
      then b29,o9 // b29,d49 by A127,A126,DIRAF:40;
      then LIN b29,o9,d49 by AFF_1:def 1;
      then LIN o9,d49,b29 by AFF_1:6;
      then
A128: o9,d49 // o9,b29 by AFF_1:def 1;
      o9,a39 // o9,d49 by A82,ANALMETR:36;
      then o9,d49 // o9,a39 by AFF_1:4;
      then o9,b29 // o9,a39 by A114,A128,DIRAF:40;
      then LIN o9,b29,a39 by AFF_1:def 1;
      then LIN b29,o9,a39 by AFF_1:6;
      then
A129: b29,o9 // b29,a39 by AFF_1:def 1;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b29,o9,b39 by AFF_1:6;
      then b29,o9 // b29,b39 by AFF_1:def 1;
      then b29,a39 // b29,b39 by A3,A11,A129,DIRAF:40;
      then LIN b29,a39,b39 by AFF_1:def 1;
      hence contradiction by A40,ANALMETR:40;
    end;
A130: not d4 in A
    proof
      assume d4 in A;
      then d19,d49 // d19,d39 by A103,A104,A105,AFF_1:39,41;
      then LIN d19,d49,d39 by AFF_1:def 1;
      hence contradiction by A117,ANALMETR:40;
    end;
A131: d2,d3 _|_ b3,a3 by A61,A62,A66,ANALMETR:62;
    take d4;
    LIN o9,b39,d19 by A31,ANALMETR:40;
    then
A132: o9,b39 // o9,d19 by AFF_1:def 1;
    LIN o9,a39,a19 by A45,ANALMETR:40;
    then
A133: LIN d29,d49,a19 by A7,A12,A58,A84,AFF_1:8;
    then consider K9 such that
A134: K9 is being_line and
A135: d29 in K9 and
A136: d49 in K9 and
A137: a19 in K9 by AFF_1:21;
    reconsider K=K9 as Subset of X;
    LIN o9,a39,a39 by AFF_1:7;
    then
A138: a3 in K by A7,A12,A58,A84,A100,A134,A135,A136,AFF_1:8,25;
    a39,a19 // a39,a29 by A4,A5,A6,A43,AFF_1:39,41;
    then a3,a1 // a3,a2 by ANALMETR:36;
    then
A139: LIN a3,a1,a2 by ANALMETR:def 10;
A140: LIN d1,d3,b3 & LIN d1,d3,b1 & LIN d2,d4,a1 & LIN d2,d4,a2
    proof
A141: b39<>b29
      proof
        assume b39=b29;
        then LIN b29,a39,b39 by AFF_1:7;
        hence contradiction by A40,ANALMETR:40;
      end;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then LIN b39,b29,o9 by AFF_1:6;
      then
A142: b39,b29 // b39,o9 by AFF_1:def 1;
      LIN b29,b39,b19 by A42,ANALMETR:40;
      then LIN b39,b29,b19 by AFF_1:6;
      then b39,b29 // b39,b19 by AFF_1:def 1;
      then b39,b19 // b39,o9 by A142,A141,DIRAF:40;
      then LIN b39,b19,o9 by AFF_1:def 1;
      then
A143: LIN o9,b39,b19 by AFF_1:6;
A144: LIN o9,b39,d39 by A70,ANALMETR:40;
      LIN o9,b39,d19 by A31,ANALMETR:40;
      then
A145: LIN d19,d39,b19 by A17,A143,A144,AFF_1:8;
      reconsider o9=o,a19=a1,a29=a2,a39=a3,d29=d2,d49=d4
       as Element of the AffinStruct of X;
A146: a39<>a19
      proof
        assume a19=a39;
        then LIN a39,a19,b29 by AFF_1:7;
        hence contradiction by A46,ANALMETR:40;
      end;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a39,a19,o9 by AFF_1:6;
      then
A147: a39,a19 // a39,o9 by AFF_1:def 1;
      LIN a39,a19,a29 by A139,ANALMETR:40;
      then a39,a19 // a39,a29 by AFF_1:def 1;
      then a39,a29 // a39,o9 by A147,A146,DIRAF:40;
      then LIN a39,a29,o9 by AFF_1:def 1;
      then
A148: LIN o9,a39,a29 by AFF_1:6;
A149: LIN o9,a39,d49 by A83,ANALMETR:40;
      LIN o9,a39,d29 by A57,ANALMETR:40;
      then LIN d29,d49,a29 by A7,A12,A148,A149,AFF_1:8;
      hence thesis by A102,A133,A145,ANALMETR:40;
    end;
    LIN o9,b39,d39 by A70,ANALMETR:40;
    then o9,b39 // o9,d39 by AFF_1:def 1;
    then o9,d19 // o9,d39 by A17,A132,DIRAF:40;
    then LIN o9,d19,d39 by AFF_1:def 1;
    then LIN d19,o9,d39 by AFF_1:6;
    then d19,o9 // d19,d39 by AFF_1:def 1;
    then o9,d19 // d19,d39 by AFF_1:4;
    then
A150: o,d1 // d1,d3 by ANALMETR:36;
    o,b3 // o,d1 by A132,ANALMETR:36;
    then o,d1 _|_ o,a3 by A17,A48,ANALMETR:62;
    then
A151: o,a3 _|_ d1,d3 by A26,A150,ANALMETR:62;
    LIN o9,a39,d29 by A57,ANALMETR:40;
    then
A152: o9,a39 // o9,d29 by AFF_1:def 1;
    then o,a3 // o,d2 by ANALMETR:36;
    then
A153: o,d2 _|_ d1,d3 by A7,A12,A151,ANALMETR:62;
A154: not d2 in A
    proof
      assume d2 in A;
      then d19,d29 // d19,d39 by A103,A104,A105,AFF_1:39,41;
      then LIN d19,d29,d39 by AFF_1:def 1;
      hence contradiction by A91,ANALMETR:40;
    end;
    LIN o9,a39,d49 by A83,ANALMETR:40;
    then o9,a39 // o9,d49 by AFF_1:def 1;
    then o9,d29 // o9,d49 by A7,A12,A152,DIRAF:40;
    then LIN o9,d29,d49 by AFF_1:def 1;
    then LIN d29,o9,d49 by AFF_1:6;
    then d29,o9 // d29,d49 by AFF_1:def 1;
    then o9,d29 // d29,d49 by AFF_1:4;
    then o,d2 // d2,d4 by ANALMETR:36;
    then
A155: d1,d3 _|_ d2,d4 by A87,A153,ANALMETR:62;
    K9=Line(d29,d49) by A100,A134,A135,A136,AFF_1:24;
    then K=Line(d2,d4) by ANALMETR:41;
    then
A156: A _|_ K by A155,A100,A107,A108,ANALMETR:45;
    reconsider d19=d1,d29=d2,d39=d3,d49=d4,b39=b3,a19=a1,b19=b1,a29=a2 as
    Element of the AffinStruct of X;
    LIN d19,d39,b39 by A140,ANALMETR:40;
    then consider A9 such that
A157: A9 is being_line and
A158: d19 in A9 and
A159: d39 in A9 and
A160: b39 in A9 by AFF_1:21;
A161: d19<>d39
    proof
      assume d19=d39;
      then LIN d19,d29,d39 by AFF_1:7;
      hence contradiction by A91,ANALMETR:40;
    end;
    reconsider A=A9 as Subset of X;
    LIN d19,d39,b19 by A140,ANALMETR:40;
    then
A162: b1 in A by A157,A158,A159,A161,AFF_1:25;
A163: not d2 in A
    proof
      assume d2 in A;
      then d19,d29 // d19,d39 by A157,A158,A159,AFF_1:39,41;
      then d1,d2 // d1,d3 by ANALMETR:36;
      hence contradiction by A91,ANALMETR:def 10;
    end;
A164: d1<>d4
    proof
      LIN o9,a39,d49 by A83,ANALMETR:40;
      then
A165: LIN o9,d49,a39 by AFF_1:6;
      LIN o9,d49,o9 by AFF_1:7;
      then
A166: o9,d49 // o9,a39 by A165,AFF_1:10;
A167: LIN o9,b39,o9 by AFF_1:7;
      LIN o9,b29,b39 by A35,ANALMETR:40;
      then
A168: LIN o9,b39,b29 by AFF_1:6;
      LIN o9,b39,d19 by A31,ANALMETR:40;
      then LIN o9,d19,b29 by A17,A167,A168,AFF_1:8;
      then
A169: o9,d19 // o9,b29 by AFF_1:def 1;
      assume d1=d4;
      then o9,a39 // o9,b29 by A114,A169,A166,DIRAF:40;
      then LIN o9,a39,b29 by AFF_1:def 1;
      then LIN a39,b29,o9 by AFF_1:6;
      then a39,b29 // a39,o9 by AFF_1:def 1;
      then
A170: a39,o9 // a39,b29 by AFF_1:4;
      LIN o9,a39,a19 by A45,ANALMETR:40;
      then LIN a39,o9,a19 by AFF_1:6;
      then a39,o9 // a39,a19 by AFF_1:def 1;
      then a39,b29 // a39,a19 by A7,A12,A170,DIRAF:40;
      then LIN a39,b29,a19 by AFF_1:def 1;
      then LIN a39,a19,b29 by AFF_1:6;
      hence contradiction by A46,ANALMETR:40;
    end;
A171: not d4 in A
    proof
      assume d4 in A;
      then d19,d49 // d19,d39 by A157,A158,A159,AFF_1:39,41;
      then d1,d4 // d1,d3 by ANALMETR:36;
      hence contradiction by A117,ANALMETR:def 10;
    end;
    LIN d29,d49,a19 by A140,ANALMETR:40;
    then consider K9 such that
A172: K9 is being_line and
A173: d29 in K9 and
A174: d49 in K9 and
A175: a19 in K9 by AFF_1:21;
    reconsider K=K9 as Subset of X;
    LIN d29,d49,a29 by A140,ANALMETR:40;
    then
A176: a2 in K by A100,A172,A173,A174,AFF_1:25;
    K9=Line(d29,d49) by A100,A172,A173,A174,AFF_1:24;
    then
A177: K=Line(d2,d4) by ANALMETR:41;
    A9=Line(d19,d39) by A157,A158,A159,A161,AFF_1:24;
    then A=Line(d1,d3) by ANALMETR:41;
    then A _|_ K by A155,A100,A161,A177,ANALMETR:45;
    then
A178: d1,d4 _|_ b3,a2 by A1,A68,A86,A55,A158,A159,A160,A173,A174,A175,A163,A171
,A162,A176;
    d1,d2 _|_ a1,b3 by A27,A28,A53,ANALMETR:62;
    then d1,d4 _|_ a1,b2 by A2,A131,A81,A104,A105,A106,A135,A136,A137,A154,A130
,A156,A110,A138;
    then a1,b2 // b3,a2 or d1=d4 by A178,ANALMETR:63;
    then a19,b29 // b39,a29 by A164,ANALMETR:36;
    then a19,b29 // a29,b39 by AFF_1:4;
    hence thesis by ANALMETR:36;
  end;
  now
A179: now
      o9,b29 // o9,b39 by A7,A9,A10,A20,AFF_1:39,41;
      then
A180: LIN o9,b29,b39 by AFF_1:def 1;
      assume
A181: a1=a2;
      a1<>b1
      proof
        o9,a19 // o9,a39 by A3,A4,A6,A21,AFF_1:39,41;
        then
A182:   LIN o9,a19,a39 by AFF_1:def 1;
        assume a1=b1;
        hence contradiction by A7,A8,A12,A14,A20,A182,AFF_1:25;
      end;
      then a3,b2 // a3,b3 by A18,A19,A181,ANALMETR:60;
      then a39,b29 // a39,b39 by ANALMETR:36;
      then b29=b39 by A3,A6,A7,A11,A12,A21,A180,AFF_1:14,25;
      then a19,b29 // a29,b39 by A181,AFF_1:2;
      hence thesis by ANALMETR:36;
    end;
A183: now
A184: not LIN o9,a39,b19
      proof
        assume LIN o9,a39,b19;
        then LIN o9,b19,a39 by AFF_1:6;
        hence contradiction by A7,A8,A12,A16,A20,AFF_1:25;
      end;
      o9,b19 // o9,b39 by A7,A8,A10,A20,AFF_1:39,41;
      then
A185: LIN o9,b19,b39 by AFF_1:def 1;
      assume
A186: a1=a3;
      then a3,b1 // a3,b3 by A19,ANALMETR:59;
      then a39,b19 // a39,b39 by ANALMETR:36;
      hence thesis by A18,A186,A184,A185,AFF_1:14;
    end;
A187: now
A188: not LIN o9,a39,b19
      proof
        assume LIN o9,a39,b19;
        then LIN o9,b19,a39 by AFF_1:6;
        hence contradiction by A7,A8,A12,A16,A20,AFF_1:25;
      end;
      o9,b19 // o9,b29 by A7,A8,A9,A20,AFF_1:39,41;
      then
A189: LIN o9,b19,b29 by AFF_1:def 1;
      assume
A190: a2=a3;
      then a3,b1 // a3,b2 by A18,ANALMETR:59;
      then a39,b19 // a39,b29 by ANALMETR:36;
      then a2,b3 // a1,b2 by A19,A190,A188,A189,AFF_1:14;
      hence thesis by ANALMETR:59;
    end;
    assume a1=a2 or a2=a3 or a1=a3;
    hence thesis by A179,A187,A183;
  end;
  hence thesis by A22;
end;
