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_3H implies X is satisfying_OPAP
proof
  assume
A1: X is satisfying_3H;
  let o,a1,a2,a3,b1,b2,b3,M,N;
  assume that
A2: o in M and
A3: a1 in M and
A4: a2 in M and
A5: a3 in M and
A6: o in N and
A7: b1 in N and
A8: b2 in N and
A9: b3 in N and
A10: not b2 in M and
A11: not a3 in N and
A12: M _|_ N and
A13: o<>a1 and
A14: o<>a2 and
  o<>a3 and
A15: o<>b1 and
  o<>b2 and
A16: o<>b3 and
A17: a3,b2 // a2,b1 and
A18: 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 A12,ANALMETR:44;
  then
A19: N9 is being_line by ANALMETR:43;
  M is being_line by A12,ANALMETR:44;
  then
A20: M9 is being_line by ANALMETR:43;
A21: now
A22: not LIN o9,b19,a19
    proof
      o9,a19 // o9,a39 by A2,A3,A5,A20,AFF_1:39,41;
      then
A23:  LIN o9,a19,a39 by AFF_1:def 1;
      assume LIN o9,b19,a19;
      then a19 in N9 by A6,A7,A15,A19,AFF_1:25;
      hence contradiction by A6,A11,A13,A19,A23,AFF_1:25;
    end;
A24: b1,a2 // a3,b2 by A17,ANALMETR:59;
    o9,a19 // o9,a29 by A2,A3,A4,A20,AFF_1:39,41;
    then
A25: LIN o9,a19,a29 by AFF_1:def 1;
    assume
A26: b2=b3;
    then b1,a1 // a3,b2 by A18,ANALMETR:59;
    then b1,a1 // b1,a2 by A5,A10,A24,ANALMETR:60;
    then b19,a19 // b19,a29 by ANALMETR:36;
    then a19=a29 by A22,A25,AFF_1:14;
    then a19,b29 // a29,b39 by A26,AFF_1:2;
    hence thesis by ANALMETR:36;
  end;
A27: LIN a,b,c implies LIN a,c,b & LIN b,a,c & LIN b,c,a & LIN c,a,b & LIN c
  ,b,a
  proof
    reconsider a9=a,b9=b,c9 = c as Element of the AffinStruct of X;
    assume LIN a,b,c;
    then
A28: LIN a9,b9,c9 by ANALMETR:40;
    then
A29: LIN b9,a9,c9 by AFF_1:6;
A30: LIN c9,a9,b9 by A28,AFF_1:6;
A31: LIN b9,c9,a9 by A28,AFF_1:6;
A32: LIN c9,b9,a9 by A28,AFF_1:6;
    LIN a9,c9,b9 by A28,AFF_1:6;
    hence thesis by A29,A31,A30,A32,ANALMETR:40;
  end;
A33: now
    o9,a39 // o9,a19 by A2,A3,A5,A20,AFF_1:39,41;
    then LIN o9,a39,a19 by AFF_1:def 1;
    then
A34: LIN o,a3,a1 by ANALMETR:40;
    a39,a19 // a39,a29 by A3,A4,A5,A20,AFF_1:39,41;
    then LIN a39,a19,a29 by AFF_1:def 1;
    then
A35: LIN a3,a1,a2 by ANALMETR:40;
    o9,b29 // o9,b39 by A6,A8,A9,A19,AFF_1:39,41;
    then LIN o9,b29,b39 by AFF_1:def 1;
    then
A36: LIN o,b2,b3 by ANALMETR:40;
    assume that
A37: a1<>a3 and
A38: b2<>b3;
A39: not LIN a3,a1,b2 & not LIN b2,a3,b3
    proof
      assume LIN a3,a1,b2 or LIN b2,a3,b3;
      then LIN a39,a19,b29 or LIN b29,a39,b39 by ANALMETR:40;
      then LIN a39,a19,b29 or LIN b29,b39,a39 by AFF_1:6;
      hence contradiction by A3,A5,A8,A9,A10,A11,A20,A19,A37,A38,AFF_1:25;
    end;
    b29,b39 // b29,b19 by A7,A8,A9,A19,AFF_1:39,41;
    then LIN b29,b39, b19 by AFF_1:def 1;
    then
A40: LIN b2,b3,b1 by ANALMETR:40;
A41: now
      assume
A42:  b2<>b1;
      not LIN a2,a3,b3
      proof
A43:    a3<>a2
        proof
          assume a3=a2;
          then LIN a3,b2,b1 by A17,ANALMETR:def 10;
          then LIN b2,b1,a3 by A27;
          then
A44:      b2,b1 // b2,a3 by ANALMETR:def 10;
          LIN b2,b1,b3 by A27,A40;
          then b2,b1 // b2,b3 by ANALMETR:def 10;
          then b2,a3 // b2,b3 by A42,A44,ANALMETR:60;
          hence contradiction by A39,ANALMETR:def 10;
        end;
        LIN a3,a1,o by A27,A34;
        then
A45:    a3,a1 // a3,o by ANALMETR:def 10;
A46:    a3<>a1
        proof
          assume a3 = a1;
          then LIN a39,a19,b29 by AFF_1:7;
          hence contradiction by A39,ANALMETR:40;
        end;
        assume LIN a2,a3,b3;
        then LIN a3,a2,b3 by A27;
        then
A47:    a3,a2 // a3,b3 by ANALMETR:def 10;
        LIN a3,a2,a1 by A27,A35;
        then a3,a2 // a3,a1 by ANALMETR:def 10;
        then a3,a1 // a3,b3 by A47,A43,ANALMETR:60;
        then a3,o // a3,b3 by A46,A45,ANALMETR:60;
        then a39,o9 // a39,b39 by ANALMETR:36;
        then
A48:    a39,b39 // a39,o9 by AFF_1:4;
        LIN b2,b3,o by A27,A36;
        then
A49:    LIN b29,b39,o9 by ANALMETR:40;
        not LIN b29,a39,b39 by A39,ANALMETR:40;
        hence contradiction by A16,A48,A49,AFF_1:14;
      end;
      then consider c1 such that
A50:  c1,a2 _|_ a3,b3 and
A51:  c1,a3 _|_ a2,b3 and
A52:  c1,b3 _|_ a2,a3 by A1,CONAFFM:def 3;
      reconsider c19 = c1 as Element of the AffinStruct of X;
A53:  now
A54:    a2<>a3
        proof
A55:      not LIN o9,a39,b19
          proof
            assume LIN o9,a39,b19;
            then LIN o9,b19,a39 by AFF_1:6;
            hence contradiction by A6,A7,A11,A15,A19,AFF_1:25;
          end;
          assume a2=a3;
          then a39,b29 // a39,b19 by A17,ANALMETR:36;
          then
A56:      a39,b19 // a39,b29 by AFF_1:4;
          o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
          then LIN o9,b19,b29 by AFF_1:def 1;
          hence contradiction by A42,A55,A56,AFF_1:14;
        end;
        a2,a3 _|_ b2,b3 by A4,A5,A8,A9,A12,ANALMETR:56;
        then b2,b3 // c1,b3 by A52,A54,ANALMETR:63;
        then b3,b2 // b3,c1 by ANALMETR:59;
        then LIN b3,b2,c1 by ANALMETR:def 10;
        then LIN b39,b29,c19 by ANALMETR:40;
        then
A57:    c1 in N by A8,A9,A19,A38,AFF_1:25;
A58:    not LIN o9,a29,c19
        proof
A59:      o9<>c19
          proof
            assume
A60:        o9=c19;
            o,a2 _|_ b3,b2 by A2,A4,A8,A9,A12,ANALMETR:56;
            then b3,b2 // a3,b3 by A14,A50,A60,ANALMETR:63;
            then b3,b2 // b3,a3 by ANALMETR:59;
            then LIN b3,b2,a3 by ANALMETR:def 10;
            then LIN b39,b29,a39 by ANALMETR:40;
            hence contradiction by A8,A9,A11,A19,A38,AFF_1:25;
          end;
          o9,a29 // o9,a39 by A2,A4,A5,A20,AFF_1:39,41;
          then
A61:      LIN o9,a29,a39 by AFF_1:def 1;
          assume LIN o9,a29,c19;
          then LIN o9,c19,a29 by AFF_1:6;
          then a29 in N9 by A6,A19,A57,A59,AFF_1:25;
          hence contradiction by A6,A11,A14,A19,A61,AFF_1:25;
        end;
A62:    a1<>b1
        proof
          o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
          then
A63:      LIN o9,b19,b29 by AFF_1:def 1;
          assume a1=b1;
          hence contradiction by A2,A3,A10,A13,A20,A63,AFF_1:25;
        end;
        assume
A64:    a2<>a1;
A65:    a1<>b1
        proof
          LIN a1,a2,a3 by A27,A35;
          then a1,a2 // a1,a3 by ANALMETR:def 10;
          then
A66:      a2,a1 // a3,a1 by ANALMETR:59;
          assume a1=b1;
          then a3,a1 // a3,b2 by A17,A64,A66,ANALMETR:60;
          hence contradiction by A39,ANALMETR:def 10;
        end;
        not LIN a2,a1,b1
        proof
          assume LIN a2,a1,b1;
          then LIN b1,a1,a2 by A27;
          then b1,a1 // b1,a2 by ANALMETR:def 10;
          then a1,b1 // a2,b1 by ANALMETR:59;
          then
A67:      a2,b1 // a3,b3 by A18,A65,ANALMETR:60;
          a2<>b1
          proof
            o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
            then
A68:        LIN o9,b19,b29 by AFF_1:def 1;
            assume a2=b1;
            hence contradiction by A2,A4,A10,A14,A20,A68,AFF_1:25;
          end;
          then a3,b3 // a3,b2 by A17,A67,ANALMETR:60;
          then LIN a3,b3,b2 by ANALMETR:def 10;
          hence contradiction by A27,A39;
        end;
        then consider c2 such that
A69:    c2,a2 _|_ a1,b1 and
A70:    c2,a1 _|_ a2,b1 and
A71:    c2,b1 _|_ a2,a1 by A1,CONAFFM:def 3;
        reconsider c29 = c2 as Element of the AffinStruct of X;
        a1,b1 _|_ c1,a2 by A9,A11,A18,A50,ANALMETR:62;
        then c2,a2 // c1,a2 by A69,A62,ANALMETR:63;
        then a2,c1 // a2,c2 by ANALMETR:59;
        then
A72:    a29,c19 // a29,c29 by ANALMETR:36;
        a2,a1 _|_ b1,b2 by A3,A4,A7,A8,A12,ANALMETR:56;
        then b1,b2 // c2,b1 by A64,A71,ANALMETR:63;
        then b1,b2 // b1,c2 by ANALMETR:59;
        then LIN b1,b2,c2 by ANALMETR:def 10;
        then LIN b19,b29,c29 by ANALMETR:40;
        then
A73:    c29 in N9 by A7,A8,A19,A42,AFF_1:25;
A74:    not LIN o9,a19,c29
        proof
A75:      o9<>c29
          proof
A76:        a2<>b1
            proof
              o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
              then
A77:          LIN o9,b19,b29 by AFF_1:def 1;
              assume a2=b1;
              hence contradiction by A2,A4,A10,A14,A20,A77,AFF_1:25;
            end;
            assume o9=c29;
            then
A78:        o,a1 _|_ a3,b2 by A17,A70,A76,ANALMETR:62;
            o,a1 _|_ b3,b2 by A2,A3,A8,A9,A12,ANALMETR:56;
            then b3,b2 // a3,b2 by A13,A78,ANALMETR:63;
            then b2,b3 // b2,a3 by ANALMETR:59;
            then LIN b2,b3,a3 by ANALMETR:def 10;
            then LIN b29,b39,a39 by ANALMETR:40;
            hence contradiction by A8,A9,A11,A19,A38,AFF_1:25;
          end;
          o9,a19 // o9,a39 by A2,A3,A5,A20,AFF_1:39,41;
          then
A79:      LIN o9,a19,a39 by AFF_1:def 1;
          assume LIN o9,a19,c29;
          then LIN o9,c29,a19 by AFF_1:6;
          then a19 in N9 by A6,A19,A73,A75,AFF_1:25;
          hence contradiction by A6,A11,A13,A19,A79,AFF_1:25;
        end;
        not LIN a3,a1,b2
        proof
          assume LIN a3,a1,b2;
          then LIN a39,a19,b29 by ANALMETR:40;
          hence contradiction by A3,A5,A10,A20,A37,AFF_1:25;
        end;
        then consider c3 such that
A80:    c3,a3 _|_ a1,b2 and
A81:    c3,a1 _|_ a3,b2 and
A82:    c3,b2 _|_ a3,a1 by A1,CONAFFM:def 3;
        reconsider c39 = c3 as Element of the AffinStruct of X;
        a2<>b1
        proof
          o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
          then
A83:      LIN o9,b19,b29 by AFF_1:def 1;
          assume a2=b1;
          hence contradiction by A2,A4,A10,A14,A20,A83,AFF_1:25;
        end;
        then a3,b2 _|_ c2,a1 by A17,A70,ANALMETR:62;
        then c2,a1 // c3,a1 by A5,A10,A81,ANALMETR:63;
        then a1,c2 // a1,c3 by ANALMETR:59;
        then
A84:    a19,c29 // a19,c39 by ANALMETR:36;
        a2,a1 _|_ b2,b1 by A3,A4,A7,A8,A12,ANALMETR:56;
        then b2,b1 // c2,b1 by A64,A71,ANALMETR:63;
        then b1,b2 // b1,c2 by ANALMETR:59;
        then LIN b1,b2,c2 by ANALMETR:def 10;
        then LIN b19,b29,c29 by ANALMETR:40;
        then c2 in N by A7,A8,A19,A42,AFF_1:25;
        then o9,c19 // o9,c29 by A6,A19,A57,AFF_1:39,41;
        then LIN o9,c19,c29 by AFF_1:def 1;
        then
A85:    c1 = c2 by A58,A72,AFF_1:14;
A86:    c1<>a3
        proof
A87:      a2<>a3
          proof
A88:        not LIN o9,a39,b19
            proof
              assume LIN o9,a39,b19;
              then LIN o9,b19,a39 by AFF_1:6;
              hence contradiction by A6,A7,A11,A15,A19,AFF_1:25;
            end;
            assume a2=a3;
            then a39,b29 // a39,b19 by A17,ANALMETR:36;
            then
A89:        a39,b19 // a39,b29 by AFF_1:4;
            o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
            then LIN o9,b19,b29 by AFF_1:def 1;
            hence contradiction by A42,A88,A89,AFF_1:14;
          end;
          assume
A90:      c1=a3;
          a2,a3 _|_ b2,b3 by A4,A5,A8,A9,A12,ANALMETR:56;
          then a3,b3 // b2,b3 by A52,A90,A87,ANALMETR:63;
          then b3,b2 // b3,a3 by ANALMETR:59;
          then LIN b3,b2,a3 by ANALMETR:def 10;
          then LIN b39,b29,a39 by ANALMETR:40;
          hence contradiction by A8,A9,A11,A19,A38,AFF_1:25;
        end;
        a3,a1 _|_ b2,b3 by A3,A5,A8,A9,A12,ANALMETR:56;
        then c3,b2 // b2,b3 by A37,A82,ANALMETR:63;
        then b2,b3 // b2,c3 by ANALMETR:59;
        then LIN b2,b3,c3 by ANALMETR:def 10;
        then LIN b29,b39,c39 by ANALMETR:40;
        then c39 in N9 by A8,A9,A19,A38,AFF_1:25;
        then o9,c29 // o9,c39 by A6,A19,A73,AFF_1:39,41;
        then LIN o9,c29,c39 by AFF_1:def 1;
        then c2 = c3 by A74,A84,AFF_1:14;
        hence thesis by A51,A85,A80,A86,ANALMETR:63;
      end;
      now
        o9,b29 // o9,b39 by A6,A8,A9,A19,AFF_1:39,41;
        then
A91:    LIN o9,b29,b39 by AFF_1:def 1;
        assume
A92:    a2=a1;
        a1<>b1
        proof
          o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
          then
A93:      LIN o9,b19,b29 by AFF_1:def 1;
          assume a1=b1;
          hence contradiction by A2,A3,A10,A13,A20,A93,AFF_1:25;
        end;
        then a3,b2 // a3,b3 by A17,A18,A92,ANALMETR:60;
        then a39,b29 // a39,b39 by ANALMETR:36;
        then b29=b39 by A2,A5,A6,A10,A11,A20,A91,AFF_1:14,25;
        then a19,b29 // a29,b39 by A92,AFF_1:2;
        hence thesis by ANALMETR:36;
      end;
      hence thesis by A53;
    end;
    now
A94:  not LIN o9,b29,a39
      proof
        assume LIN o9,b29,a39;
        then LIN o,b2,a3 by ANALMETR:40;
        then LIN b2,o,a3 by A27;
        then
A95:    b2,o // b2,a3 by ANALMETR:def 10;
        LIN b2,o,b3 by A27,A36;
        then b2,o // b2,b3 by ANALMETR:def 10;
        then b2,a3 // b2,b3 by A2,A10,A95,ANALMETR:60;
        hence contradiction by A39,ANALMETR:def 10;
      end;
      LIN a3,a1,o by A27,A34;
      then
A96:  a3,a1 // a3,o by ANALMETR:def 10;
A97:  a3<>a1
      proof
        assume a3=a1;
        then LIN a39,a19,b29 by AFF_1:7;
        hence contradiction by A39,ANALMETR:40;
      end;
      a3,a1 // a3,a2 by A35,ANALMETR:def 10;
      then a3,o // a3,a2 by A96,A97,ANALMETR:60;
      then LIN a3,o,a2 by ANALMETR:def 10;
      then LIN o,a3,a2 by A27;
      then
A98:  LIN o9,a39,a29 by ANALMETR:40;
      assume
A99:  b2=b1;
      then b2,a3 // b2,a2 by A17,ANALMETR:59;
      then b29,a39 // b29,a29 by ANALMETR:36;
      then a39=a29 by A98,A94,AFF_1:14;
      hence thesis by A18,A99,ANALMETR:59;
    end;
    hence thesis by A41;
  end;
  now
A100: not LIN o9,a19,b19
    proof
      o9,b19 // o9,b29 by A6,A7,A8,A19,AFF_1:39,41;
      then
A101: LIN o9,b19,b29 by AFF_1:def 1;
      assume LIN o9,a19,b19;
      then b19 in M9 by A2,A3,A13,A20,AFF_1:25;
      hence contradiction by A2,A10,A15,A20,A101,AFF_1:25;
    end;
    o9,b19 // o9,b39 by A6,A7,A9,A19,AFF_1:39,41;
    then
A102: LIN o9,b19,b39 by AFF_1:def 1;
    assume
A103: a1=a3;
    then a1,b1 // a1,b3 by A18,ANALMETR:59;
    then a19,b19 // a19,b39 by ANALMETR:36;
    hence thesis by A17,A103,A100,A102,AFF_1:14;
  end;
  hence thesis by A21,A33;
end;
