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_MH2 implies X is satisfying_OSCH
proof
  assume
A1: X is satisfying_MH2;
  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
      ex d39 be Element of the AffinStruct of X st o9<>d39 & d39 in N9
by A21,AFF_1:20;
      then consider d3 such that
A28:  d3 in N and
A29:  d3 <>o;
      reconsider d39=d3 as Element of the AffinStruct of X;
      consider e2 such that
A30:  a3,a2 _|_ d3,e2 and
A31:  d3<>e2 by ANALMETR:def 9;
      consider e1 such that
A32:  a3,a1 // a3,e1 and
A33:  a3<>e1 by ANALMETR:39;
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      assume
A34:  b1<>b3;
A35:  a1<>a3 & a2<>a4
      proof
        assume a1=a3 or a2=a4;
        then a2,a1 // b3,b2 or a1,a4 // b2,b1 by A18,A19,ANALMETR:59;
        then b3,b2 // b2,b1 or b2,b1 // b1,b4 by A3,A11,A12,A19,A20,ANALMETR:60
;
        then b2,b3 // b2,b1 or b1,b2 // b1,b4 by ANALMETR:59;
        then LIN b2,b3,b1 or LIN b1,b2,b4 by ANALMETR:def 10;
        then LIN b29,b39,b19 or LIN b19,b29,b49 by ANALMETR:40;
        then LIN b19,b39,b29 or LIN b29,b49,b19 by AFF_1:6;
        hence contradiction by A5,A6,A9,A10,A13,A17,A22,A21,A26,A34,AFF_1:25;
      end;
      not a39,e19 // d39,e29
      proof
        assume a39,e19 // d39,e29;
        then a3,e1 // d3,e2 by ANALMETR:36;
        then a3,a1 // d3,e2 by A32,A33,ANALMETR:60;
        then
A36:    a3,a2 _|_ a3,a1 by A30,A31,ANALMETR:62;
        a3,a1 _|_ a2,a4 by A2,A3,A4,A7,A8,ANALMETR:56;
        then a3,a2 // a2,a4 by A35,A36,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,A35,AFF_1:25;
      end;
      then consider d29 be Element of the AffinStruct of X such that
A37:  LIN a39,e19,d29 and
A38:  LIN d39,e29,d29 by AFF_1:60;
      reconsider d2=d29 as Element of X;
      LIN d3,e2,d2 by A38,ANALMETR:40;
      then d3,e2 // d3,d2 by ANALMETR:def 10;
      then
A39:  a3,a2 _|_ d3,d2 by A30,A31,ANALMETR:62;
      LIN a3,e1,d2 by A37,ANALMETR:40;
      then a3,e1 // a3,d2 by ANALMETR:def 10;
      then a3,a1 // a3,d2 by A32,A33,ANALMETR:60;
      then LIN a3,a1,d2 by ANALMETR:def 10;
      then LIN a39,a19,d29 by ANALMETR:40;
      then consider d2 such that
A40:  d2 in M and
A41:  a3,a2 _|_ d3,d2 by A3,A4,A22,A35,A39,AFF_1:25;
      reconsider d29=d2 as Element of the AffinStruct of X;
      consider e1 such that
A42:  a2,a4 // a2,e1 and
A43:  a2<>e1 by ANALMETR:39;
      consider e2 such that
A44:  a2,a1 _|_ d2,e2 and
A45:  d2<>e2 by ANALMETR:def 9;
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      not a29,e19 // d29,e29
      proof
        assume a29,e19 // d29,e29;
        then a2,e1 // d2,e2 by ANALMETR:36;
        then a2,a4 // d2,e2 by A42,A43,ANALMETR:60;
        then
A46:    a2,a4 _|_ a2,a1 by A44,A45,ANALMETR:62;
        a1,a3 _|_ a2,a4 by A2,A3,A4,A7,A8,ANALMETR:56;
        then a1,a3 // a2,a1 by A35,A46,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,A35,AFF_1:25;
      end;
      then consider d19 be Element of the AffinStruct of X such that
A47:  LIN a29,e19,d19 and
A48:  LIN d29,e29,d19 by AFF_1:60;
      reconsider d1=d19 as Element of X;
A49:  b3,b2 _|_ d3,d2 by A4,A12,A18,A41,ANALMETR:62;
      LIN a2,e1,d1 by A47,ANALMETR:40;
      then a2,e1 // a2,d1 by ANALMETR:def 10;
      then a2,a4 // a2,d1 by A42,A43,ANALMETR:60;
      then LIN a2,a4,d1 by ANALMETR:def 10;
      then LIN a29,a49,d19 by ANALMETR:40;
      then
A50:  d1 in N by A7,A8,A21,A35,AFF_1:25;
      LIN d2,e2,d1 by A48,ANALMETR:40;
      then d2,e2 // d2,d1 by ANALMETR:def 10;
      then
A51:  a2,a1 _|_ d2,d1 by A44,A45,ANALMETR:62;
      then
A52:  b2,b1 _|_ d2,d1 by A3,A12,A19,ANALMETR:62;
      consider e2 such that
A53:  a1,a4 _|_ d1,e2 and
A54:  d1<>e2 by ANALMETR:39;
      consider e1 such that
A55:  a1,a3 // a1,e1 and
A56:  a1<>e1 by ANALMETR:39;
      reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
      not a19,e19 // d19,e29
      proof
        assume a19,e19 // d19,e29;
        then a1,e1 // d1,e2 by ANALMETR:36;
        then a1,a3 // d1,e2 by A55,A56,ANALMETR:60;
        then
A57:    a1,a3 _|_ a1,a4 by A53,A54,ANALMETR:62;
        a1,a3 _|_ a2,a4 by A2,A3,A4,A7,A8,ANALMETR:56;
        then a2,a4 // a1,a4 by A35,A57,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 d49 be Element of the AffinStruct of X such that
A58:  LIN a19,e19,d49 and
A59:  LIN d19,e29,d49 by AFF_1:60;
      reconsider d4=d49 as Element of X;
      LIN a1,e1,d4 by A58,ANALMETR:40;
      then a1,e1 // a1,d4 by ANALMETR:def 10;
      then a1,a3 // a1,d4 by A55,A56,ANALMETR:60;
      then LIN a1,a3,d4 by ANALMETR:def 10;
      then LIN a19,a39,d49 by ANALMETR:40;
      then
A60:  d4 in M by A3,A4,A22,A35,AFF_1:25;
      then
A61:  d3<>d4 by A2,A22,A21,A23,A24,A28,A29,AFF_1:18;
      LIN d1,e2,d4 by A59,ANALMETR:40;
      then d1,e2 // d1,d4 by ANALMETR:def 10;
      then
A62:  a1,a4 _|_ d1,d4 by A53,A54,ANALMETR:62;
      then b1,b4 _|_ d1,d4 by A3,A11,A20,ANALMETR:62;
      then
A63:  b3,b4 _|_ d3,d4 by A1,A2,A5,A6,A9,A10,A13,A14,A28,A40,A50,A60,A49,A52;
      a3,a4 _|_ d3,d4 by A1,A2,A3,A4,A7,A8,A11,A12,A28,A40,A41,A50,A51,A60,A62;
      hence thesis by A63,A61,ANALMETR:63;
    end;
    now
      o9,a19 // o9,a39 by A3,A4,A22,A23,AFF_1:39,41;
      then
A64:  LIN o9,a19,a39 by AFF_1:def 1;
      assume
A65:  b1=b3;
      then a2,a1 // b3,b2 by A19,ANALMETR:59;
      then a2,a1 // a3,a2 by A6,A13,A18,ANALMETR:60;
      then a2,a1 // a2,a3 by ANALMETR:59;
      then a29,a19 // a29,a39 by ANALMETR:36;
      hence thesis by A7,A12,A15,A20,A21,A23,A24,A65,A64,AFF_1:14,25;
    end;
    hence thesis by A27;
  end;
  now
    o9,a29 // o9,a49 by A7,A8,A21,A24,AFF_1:39,41;
    then
A66: LIN o9,a29,a49 by AFF_1:def 1;
    assume
A67: b2=b4;
    then a1,a4 // b2,b1 by A20,ANALMETR:59;
    then a2,a1 // a1,a4 by A5,A13,A19,ANALMETR:60;
    then a1,a2 // a1,a4 by ANALMETR:59;
    then a19,a29 // a19,a49 by ANALMETR:36;
    hence thesis by A3,A12,A15,A18,A22,A23,A24,A67,A66,AFF_1:14,25;
  end;
  hence thesis by A25;
end;
