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 implies X is satisfying_OSCH
proof
  assume
A1: X is satisfying_MH1;
  let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
  assume that
A2: M _|_ N and
A3: a1 in M and
A4: a3 in M and
A5: b1 in M and
A6: b3 in M and
A7: a2 in N and
A8: a4 in N and
A9: b2 in N and
A10: b4 in N and
A11: not a4 in M and
A12: not a2 in M and
A13: not b2 in M and
A14: not b4 in M and
A15: not a1 in N and
A16: not a3 in N and
A17: not b1 in N and
  not b3 in N and
A18: a3,a2 // b3,b2 and
A19: a2,a1 // b2,b1 and
A20: a1,a4 // b1,b4;
  reconsider M9=M,N9=N as Subset of the AffinStruct of X;
  N is being_line by A2,ANALMETR:44;
  then
A21: N9 is being_line by ANALMETR:43;
  reconsider b49=b4,b19=b1,b29=b2,b39=b3,a19=a1,a29=a2,a39=a3,a49=a4 as
  Element of the AffinStruct of X;
  M is being_line by A2,ANALMETR:44;
  then
A22: M9 is being_line by ANALMETR:43;
  not M9 // N9
  proof
    assume M9 // N9;
    then M // N by ANALMETR:46;
    hence contradiction by A2,ANALMETR:52;
  end;
  then ex o9 be Element of the AffinStruct of X st o9 in M9 & o9 in N9
    by A22,A21,AFF_1:58;
  then consider o such that
A23: o in M and
A24: o in N;
  reconsider o9=o as Element of the AffinStruct of X;
A25: now
    assume
A26: b2<>b4;
A27: now
      consider e19 be Element of the AffinStruct of X such that
A28:  o9<>e19 and
A29:  e19 in M9 by A22,AFF_1:20;
      reconsider e1=e19 as Element of X;
      ex d49 be Element of the AffinStruct of X st o9<>d49 & d49 in N9
   by A21,AFF_1:20;
      then consider d4 such that
A30:  d4 in N and
A31:  d4 <> o;
      reconsider d49=d4 as Element of the AffinStruct of X;
      consider e2 such that
A32:  a1,a4 _|_ d4,e2 and
A33:  e2<>d4 by ANALMETR:39;
      reconsider e29=e2 as Element of the AffinStruct of X;
      assume
A34:  b1<>b3;
      not o9,e19 // d49,e29
      proof
A35:    a49<>a29
        proof
          assume a49=a29;
          then a2,a1 // b1,b4 by A20,ANALMETR:59;
          then b1,b4 // b2,b1 by A3,A12,A19,ANALMETR:60;
          then b1,b4 // b1,b2 by ANALMETR:59;
          then LIN b1,b4,b2 by ANALMETR:def 10;
          then LIN b19,b49,b29 by ANALMETR:40;
          then LIN b29,b49,b19 by AFF_1:6;
          hence contradiction by A9,A10,A17,A21,A26,AFF_1:25;
        end;
A36:    a1<>a3
        proof
          assume a1=a3;
          then a3,a2 // b2,b1 by A19,ANALMETR:59;
          then b3,b2 // b2,b1 by A4,A12,A18,ANALMETR:60;
          then b2,b3 // b2,b1 by ANALMETR:59;
          then LIN b2,b3,b1 by ANALMETR:def 10;
          then LIN b29,b39,b19 by ANALMETR:40;
          then LIN b19,b39,b29 by AFF_1:6;
          hence contradiction by A5,A6,A13,A22,A34,AFF_1:25;
        end;
        assume o9,e19 // d49,e29;
        then o,e1 // d4,e2 by ANALMETR:36;
        then
A37:    o,e1 _|_ a1,a4 by A32,A33,ANALMETR:62;
        o9,e19 // a19,a39 by A3,A4,A22,A23,A29,AFF_1:39,41;
        then o,e1 // a1,a3 by ANALMETR:36;
        then
A38:    a1,a3 _|_ a1,a4 by A28,A37,ANALMETR:62;
        a1,a3 _|_ a2,a4 by A2,A3,A4,A7,A8,ANALMETR:56;
        then a1,a4 // a2,a4 by A38,A36,ANALMETR:63;
        then a4,a2 // a4,a1 by ANALMETR:59;
        then LIN a4,a2,a1 by ANALMETR:def 10;
        then LIN a49,a29,a19 by ANALMETR:40;
        hence contradiction by A7,A8,A15,A21,A35,AFF_1:25;
      end;
      then consider d19 be Element of the AffinStruct of X such that
A39:  LIN o9,e19,d19 and
A40:  LIN d49,e29,d19 by AFF_1:60;
      reconsider d1=d19 as Element of X;
      consider e19 be Element of the AffinStruct of X such that
A41:  o9<>e19 and
A42:  e19 in N9 by A21,AFF_1:20;
A43:  d1 in M by A22,A23,A28,A29,A39,AFF_1:25;
      LIN d4,e2,d1 by A40,ANALMETR:40;
      then d4,e2 // d4,d1 by ANALMETR:def 10;
      then
A44:  d1,d4 // d4,e2 by ANALMETR:59;
      then a1,a4 _|_ d1,d4 by A32,A33,ANALMETR:62;
      then
A45:  b1,b4 _|_ d1,d4 by A3,A11,A20,ANALMETR:62;
      reconsider e1=e19 as Element of X;
      consider e2 such that
A46:  a2,a1 _|_ d1,e2 and
A47:  e2<>d1 by ANALMETR:39;
      reconsider e29=e2 as Element of the AffinStruct of X;
      not o9,e19 // e29,d19
      proof
A48:    a19<>a39
        proof
          assume a19=a39;
          then a3,a2 // b2,b1 by A19,ANALMETR:59;
          then b3,b2 // b2,b1 by A4,A12,A18,ANALMETR:60;
          then b2,b3 // b2,b1 by ANALMETR:59;
          then LIN b2,b3,b1 by ANALMETR:def 10;
          then LIN b29,b39,b19 by ANALMETR:40;
          then LIN b19,b39,b29 by AFF_1:6;
          hence contradiction by A5,A6,A13,A22,A34,AFF_1:25;
        end;
        o9,e19 // a29,a49 by A7,A8,A21,A24,A42,AFF_1:39,41;
        then
A49:    o,e1 // a2,a4 by ANALMETR:36;
A50:    a2<>a4
        proof
          assume a2=a4;
          then a2,a1 // b1,b4 by A20,ANALMETR:59;
          then b1,b4 // b2,b1 by A3,A12,A19,ANALMETR:60;
          then b1,b4 // b1,b2 by ANALMETR:59;
          then LIN b1,b4,b2 by ANALMETR:def 10;
          then LIN b19,b49,b29 by ANALMETR:40;
          then LIN b29,b49,b19 by AFF_1:6;
          hence contradiction by A9,A10,A17,A21,A26,AFF_1:25;
        end;
        assume o9,e19 // e29,d19;
        then
A51:    o,e1 // e2,d1 by ANALMETR:36;
        a2,a1 _|_ e2,d1 by A46,ANALMETR:61;
        then o,e1 _|_ a2,a1 by A47,A51,ANALMETR:62;
        then
A52:    a2,a1 _|_ a2,a4 by A41,A49,ANALMETR:62;
        a3,a1 _|_ a2,a4 by A2,A3,A4,A7,A8,ANALMETR:56;
        then a3,a1 // a2,a1 by A52,A50,ANALMETR:63;
        then a1,a3 // a1,a2 by ANALMETR:59;
        then LIN a1,a3,a2 by ANALMETR:def 10;
        then LIN a19,a39,a29 by ANALMETR:40;
        hence contradiction by A3,A4,A12,A22,A48,AFF_1:25;
      end;
      then consider d29 be Element of the AffinStruct of X such that
A53:  LIN o9,e19,d29 and
A54:  LIN e29,d19,d29 by AFF_1:60;
      reconsider d2=d29 as Element of X;
A55:  d2 in N by A21,A24,A41,A42,A53,AFF_1:25;
      LIN d19,d29,e29 by A54,AFF_1:6;
      then LIN d1,d2,e2 by ANALMETR:40;
      then d1,d2 // d1,e2 by ANALMETR:def 10;
      then
A56:  d2,d1 // e2,d1 by ANALMETR:59;
A57:  a2,a1 _|_ e2,d1 by A46,ANALMETR:61;
      then
A58:  a2,a1 _|_ d2,d1 by A47,A56,ANALMETR:62;
      consider e19 be Element of the AffinStruct of X such that
A59:  o9<>e19 and
A60:  e19 in M9 by A22,AFF_1:20;
      reconsider e1=e19 as Element of X;
      consider e2 such that
A61:  a3,a2 _|_ d2,e2 and
A62:  e2<>d2 by ANALMETR:39;
      reconsider e29=e2 as Element of the AffinStruct of X;
      not o9,e19 // e29,d29
      proof
A63:    a29<>a49
        proof
          assume a29=a49;
          then a2,a1 // b1,b4 by A20,ANALMETR:59;
          then b1,b4 // b2,b1 by A3,A12,A19,ANALMETR:60;
          then b1,b4 // b1,b2 by ANALMETR:59;
          then LIN b1,b4,b2 by ANALMETR:def 10;
          then LIN b19,b49,b29 by ANALMETR:40;
          then LIN b29,b49,b19 by AFF_1:6;
          hence contradiction by A9,A10,A17,A21,A26,AFF_1:25;
        end;
        assume o9,e19 // e29,d29;
        then o,e1 // e2,d2 by ANALMETR:36;
        then o,e1 // d2,e2 by ANALMETR:59;
        then
A64:    a3,a2 _|_ o,e1 by A61,A62,ANALMETR:62;
        o,e1 _|_ a2,a4 by A2,A7,A8,A23,A60,ANALMETR:56;
        then a3,a2 // a2,a4 by A59,A64,ANALMETR:63;
        then a2,a4 // a2,a3 by ANALMETR:59;
        then LIN a2,a4,a3 by ANALMETR:def 10;
        then LIN a29,a49,a39 by ANALMETR:40;
        hence contradiction by A7,A8,A16,A21,A63,AFF_1:25;
      end;
      then consider d39 be Element of the AffinStruct of X such that
A65:  LIN o9,e19,d39 and
A66:  LIN e29,d29,d39 by AFF_1:60;
      reconsider d3=d39 as Element of X;
A67:  d3 in M by A22,A23,A59,A60,A65,AFF_1:25;
      then
A68:  d3<>d4 by A2,A22,A21,A23,A24,A30,A31,AFF_1:18;
      a2,a1 _|_ d2,d1 by A47,A56,A57,ANALMETR:62;
      then
A69:  b2,b1 _|_ d2,d1 by A3,A12,A19,ANALMETR:62;
      LIN d29,d39,e29 by A66,AFF_1:6;
      then LIN d2,d3,e2 by ANALMETR:40;
      then d2,d3 // d2,e2 by ANALMETR:def 10;
      then d3,d2 // d2,e2 by ANALMETR:59;
      then
A70:  a3,a2 _|_ d3,d2 by A61,A62,ANALMETR:62;
      then b3,b2 _|_ d3,d2 by A4,A12,A18,ANALMETR:62;
      then
A71:  b3,b4 _|_ d3,d4 by A1,A2,A5,A6,A9,A10,A13,A14,A30,A43,A55,A67,A45,A69;
      a1,a4 _|_ d1,d4 by A32,A33,A44,ANALMETR:62;
      then a3,a4 _|_ d3,d4 by A1,A2,A3,A4,A7,A8,A11,A12,A30,A43,A55,A58,A70,A67
;
      hence thesis by A68,A71,ANALMETR:63;
    end;
    now
A72:  not LIN o9,a29,a19 by A7,A12,A15,A21,A23,A24,AFF_1:48,def 1;
A73:  LIN o9,a19,a39 by A3,A4,A22,A23,AFF_1:21;
      assume
A74:  b1=b3;
      then a2,a3 // b2,b1 by A18,ANALMETR:59;
      then
A75:  a29,a39 // b29,b19 by ANALMETR:36;
      a29,a19 // b29,b19 by A19,ANALMETR:36;
      then a29,a19 // a29,a39 by A5,A13,A75,AFF_1:5;
      hence thesis by A20,A74,A72,A73,AFF_1:14;
    end;
    hence thesis by A27;
  end;
  now
A76: not LIN o9,a19,a49
    proof
      assume LIN o9,a19,a49;
      then
      ex A9 st A9 is being_line & o9 in A9 & a19 in A9 & a49 in A9 by AFF_1:21;
      hence contradiction by A3,A11,A15,A22,A23,A24,AFF_1:18;
    end;
    assume
A77: b2=b4;
    b1,b2 // a1,a2 by A19,ANALMETR:59;
    then a1,a4 // a1,a2 by A5,A13,A20,A77,ANALMETR:60;
    then
A78: a19,a49 // a19,a29 by ANALMETR:36;
    LIN o9,a49,a29 by A7,A8,A21,A24,AFF_1:21;
    hence thesis by A18,A77,A78,A76,AFF_1:14;
  end;
  hence thesis by A25;
end;
