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_OSCH & X is satisfying_TDES implies X is satisfying_SCH
proof
  assume that
A1: X is satisfying_OSCH and
A2: X is satisfying_TDES;
  let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
  assume that
A3: M is being_line and
A4: N is being_line and
A5: a1 in M and
A6: a3 in M and
A7: b1 in M and
A8: b3 in M and
A9: a2 in N and
A10: a4 in N and
A11: b2 in N and
A12: b4 in N and
A13: not a4 in M and
A14: not a2 in M and
A15: not b2 in M and
A16: not b4 in M and
A17: not a1 in N and
A18: not a3 in N and
A19: not b1 in N and
A20: not b3 in N and
A21: a3,a2 // b3,b2 and
A22: a2,a1 // b2,b1 and
A23: a1,a4 // b1,b4;
  reconsider a19=a1,a29=a2,a39=a3,a49=a4,b19=b1,b29=b2,b39=b3,b49=b4 as
  Element of the AffinStruct of X;
  reconsider M9=M,N9=N as Subset of the AffinStruct of X;
A24: M9 is being_line by A3,ANALMETR:43;
A25: N9 is being_line by A4,ANALMETR:43;
A26: a1<>b1 implies a2<>b2 & a3<>b3 & a4<>b4
  proof
    assume
A27: a1<>b1;
A28: now
      assume a4=b4;
      then a4,a1 // a4,b1 by A23,ANALMETR:59;
      then LIN a4,a1,b1 by ANALMETR:def 10;
      then LIN a49,a19,b19 by ANALMETR:40;
      then LIN a19,b19,a49 by AFF_1:6;
      hence contradiction by A5,A7,A13,A24,A27,AFF_1:25;
    end;
A29: now
      assume a2=b2;
      then LIN a2,a1,b1 by A22,ANALMETR:def 10;
      then LIN a29,a19,b19 by ANALMETR:40;
      then LIN a19,b19,a29 by AFF_1:6;
      hence contradiction by A5,A7,A14,A24,A27,AFF_1:25;
    end;
A30: now
      assume a3=b3;
      then LIN a3,a2,b2 by A21,ANALMETR:def 10;
      then LIN a39,a29,b29 by ANALMETR:40;
      then LIN a29,b29,a39 by AFF_1:6;
      hence contradiction by A9,A11,A18,A25,A29,AFF_1:25;
    end;
    assume a2=b2 or a3=b3 or a4=b4;
    hence contradiction by A29,A28,A30;
  end;
A31: now
    assume
A32: a1<>b1;
A33: now
A34:  not LIN a29,b19,b29
      proof
        assume LIN a29,b19,b29;
        then LIN a29,b29,b19 by AFF_1:6;
        hence contradiction by A9,A11,A19,A25,A26,A32,AFF_1:25;
      end;
      assume
A35:  a2=a4;
      then a2,a1 // b1,b4 by A23,ANALMETR:59;
      then b2,b1 // b1,b4 by A5,A14,A22,ANALMETR:60;
      then b1,b2 // b1,b4 by ANALMETR:59;
      then
A36:  b19,b29 // b19,b49 by ANALMETR:36;
      a29,b29 // a29,b49 by A9,A11,A12,A25,AFF_1:39,41;
      then LIN a29,b29,b49 by AFF_1:def 1;
      hence thesis by A21,A35,A34,A36,AFF_1:14;
    end;
A37: now
      assume that
A38:  a2<>a4 and
A39:  a1<>a3;
A40:  now
        consider e1 be Element of X such that
A41:    b1,b2 // b1,e1 and
A42:    b1<>e1 by ANALMETR:39;
        consider x be Element of X such that
A43:    LIN a1,a2,x and
A44:    a1<>x and
A45:    a2<>x by Th2;
        reconsider x9=x as Element of the AffinStruct of X;
        consider e2 be Element of X such that
A46:    a1,b1 // x,e2 and
A47:    x<>e2 by ANALMETR:39;
        reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
        not b19,e19 // x9,e29
        proof
          assume b19,e19 // x9,e29;
          then b1,e1 // x,e2 by ANALMETR:36;
          then b1,b2 // x,e2 by A41,A42,ANALMETR:60;
          then b1,b2 // a1,b1 by A46,A47,ANALMETR:60;
          then b1,a1 // b1,b2 by ANALMETR:59;
          then LIN b1,a1,b2 by ANALMETR:def 10;
          then LIN b19,a19,b29 by ANALMETR:40;
          hence contradiction by A5,A7,A15,A24,A32,AFF_1:25;
        end;
        then consider y9 be Element of the AffinStruct of X such that
A48:    LIN b19,e19,y9 and
A49:    LIN x9,e29,y9 by AFF_1:60;
        reconsider y=y9 as Element of X;
A50:    a1,a2 // a1,x by A43,ANALMETR:def 10;
A51:    not LIN a2,b2,x
        proof
A52:      a2<>b2
          proof
            assume a2=b2;
            then LIN a2,a1,b1 by A22,ANALMETR:def 10;
            then LIN a29,a19,b19 by ANALMETR:40;
            then LIN a19,b19,a29 by AFF_1:6;
            hence contradiction by A5,A7,A14,A24,A32,AFF_1:25;
          end;
          LIN a19,a29,x9 by A43,ANALMETR:40;
          then
A53:      LIN a29,x9,a19 by AFF_1:6;
          assume LIN a2,b2,x;
          then LIN a29,b29,x9 by ANALMETR:40;
          then x9 in N9 by A9,A11,A25,A52,AFF_1:25;
          hence contradiction by A9,A17,A25,A45,A53,AFF_1:25;
        end;
A54:    not LIN a1,b1,a4
        proof
          assume LIN a1,b1,a4;
          then LIN a19,b19,a49 by ANALMETR:40;
          hence contradiction by A5,A7,A13,A24,A32,AFF_1:25;
        end;
A55:    not LIN a1,b1,x
        proof
          assume LIN a1,b1,x;
          then LIN a19,b19,x9 by ANALMETR:40;
          then
A56:      x9 in M9 by A5,A7,A24,A32,AFF_1:25;
          LIN a19,a29,x9 by A43,ANALMETR:40;
          then LIN a19,x9,a29 by AFF_1:6;
          hence contradiction by A5,A14,A24,A44,A56,AFF_1:25;
        end;
        assume
A57:    M // N;
        then M9 // N9 by ANALMETR:46;
        then a39,b39 // a29, b29 by A6,A8,A9,A11,AFF_1:39;
        then a3,b3 // a2,b2 by ANALMETR:36;
        then
A58:    a2,b2 // a3,b3 by ANALMETR:59;
        LIN x,e2,y by A49,ANALMETR:40;
        then x,e2 // x,y by ANALMETR:def 10;
        then
A59:    a1,b1 // x,y by A46,A47,ANALMETR:60;
A60:    b1 <> y
        proof
          assume b1=y;
          then b1,a1 // b1,x by A59,ANALMETR:59;
          then LIN b1,a1,x by ANALMETR:def 10;
          then LIN b19,a19,x9 by ANALMETR:40;
          then
A61:      x9 in M9 by A5,A7,A24,A32,AFF_1:25;
          LIN a19,a29,x9 by A43,ANALMETR:40;
          then LIN a19,x9,a29 by AFF_1:6;
          hence contradiction by A5,A14,A24,A44,A61,AFF_1:25;
        end;
        LIN b1,e1,y by A48,ANALMETR:40;
        then b1,e1 // b1,y by ANALMETR:def 10;
        then
A62:    b1,b2 // b1,y by A41,A42,ANALMETR:60;
        then LIN b1,b2,y by ANALMETR:def 10;
        then LIN b19,b29,y9 by ANALMETR:40;
        then LIN y9,b19,b29 by AFF_1:6;
        then LIN y,b1,b2 by ANALMETR:40;
        then y,b1 // y,b2 by ANALMETR:def 10;
        then
A63:    b1,y // b2,y by ANALMETR:59;
        a1,a2 // b1, b2 by A22,ANALMETR:59;
        then a1,a2 // b1,y by A7,A15,A62,ANALMETR:60;
        then
A64:    a1,x // b1,y by A5,A14,A50,ANALMETR:60;
A65:    not LIN x,y,a3
        proof
A66:      x<>y
          proof
            assume x=y;
            then x,a1 // x,b1 by A64,ANALMETR:59;
            then LIN x,a1,b1 by ANALMETR:def 10;
            then LIN x9,a19,b19 by ANALMETR:40;
            then LIN a19,b19,x9 by AFF_1:6;
            then
A67:        x9 in M9 by A5,A7,A24,A32,AFF_1:25;
            LIN a19,a29,x9 by A43,ANALMETR:40;
            then LIN x9,a19, a29 by AFF_1:6;
            hence contradiction by A5,A14,A24,A44,A67,AFF_1:25;
          end;
          a19,a39 // a19,b19 by A5,A6,A7,A24,AFF_1:39,41;
          then a1,a3 // a1,b1 by ANALMETR:36;
          then
A68:      x,y // a1,a3 by A32,A59,ANALMETR:60;
          assume LIN x,y,a3;
          then x,y // x,a3 by ANALMETR:def 10;
          then x,a3 // a1,a3 by A66,A68,ANALMETR:60;
          then a3,a1 // a3,x by ANALMETR:59;
          then LIN a3,a1,x by ANALMETR:def 10;
          then LIN a39,a19,x9 by ANALMETR:40;
          then
A69:      x9 in M9 by A5,A6,A24,A39,AFF_1:25;
          LIN a19,a29,x9 by A43,ANALMETR:40;
          then LIN a19,x9, a29 by AFF_1:6;
          hence contradiction by A5,A14,A24,A44,A69,AFF_1:25;
        end;
A70:    not LIN x,y,a4
        proof
A71:      x<>y
          proof
            assume x=y;
            then x,a1 // x,b1 by A64,ANALMETR:59;
            then LIN x,a1,b1 by ANALMETR:def 10;
            then LIN x9,a19,b19 by ANALMETR:40;
            then LIN a19,b19,x9 by AFF_1:6;
            then
A72:        x9 in M9 by A5,A7,A24,A32,AFF_1:25;
            LIN a19,a29,x9 by A43,ANALMETR:40;
            then LIN x9,a19, a29 by AFF_1:6;
            hence contradiction by A5,A14,A24,A44,A72,AFF_1:25;
          end;
          M9 // N9 by A57,ANALMETR:46;
          then a19,b19 // a29,a49 by A5,A7,A9,A10,AFF_1:39;
          then a1,b1 // a2,a4 by ANALMETR:36;
          then
A73:      a2,a4 // x,y by A32,A59,ANALMETR:60;
          assume LIN x,y,a4;
          then x,y // x,a4 by ANALMETR:def 10;
          then a2,a4 // x,a4 by A71,A73,ANALMETR:60;
          then a4,a2 // a4,x by ANALMETR:59;
          then LIN a4,a2,x by ANALMETR:def 10;
          then LIN a49,a29,x9 by ANALMETR:40;
          then
A74:      x9 in N9 by A9,A10,A25,A38,AFF_1:25;
          LIN a19,a29,x9 by A43,ANALMETR:40;
          then LIN a29,x9, a19 by AFF_1:6;
          hence contradiction by A9,A17,A25,A45,A74,AFF_1:25;
        end;
A75:    not LIN a2,b2,a3
        proof
A76:      a2<>b2
          proof
            assume a2=b2;
            then LIN a2,a1,b1 by A22,ANALMETR:def 10;
            then LIN a29,a19,b19 by ANALMETR:40;
            then LIN a19,b19,a29 by AFF_1:6;
            hence contradiction by A5,A7,A14,A24,A32,AFF_1:25;
          end;
          assume LIN a2,b2,a3;
          then LIN a29,b29,a39 by ANALMETR:40;
          hence contradiction by A9,A11,A18,A25,A76,AFF_1:25;
        end;
A77:    X is satisfying_TDES implies X is satisfying_des
        proof
          assume X is satisfying_TDES;
          then the AffinStruct of X is Moufangian by Th7;
          then the AffinStruct of X is translational by AFF_2:14;
          hence thesis by Th8;
        end;
        LIN a19,a29,x9 by A43,ANALMETR:40;
        then LIN x9,a19, a29 by AFF_1:6;
        then LIN x,a1,a2 by ANALMETR:40;
        then x,a1 // x,a2 by ANALMETR:def 10;
        then a1,x // a2,x by ANALMETR:59;
        then a2,x // b1,y by A44,A64,ANALMETR:60;
        then
A78:    a2,x // b2,y by A60,A63,ANALMETR:60;
        a19,b19 // a39,b39 by A5,A6,A7,A8,A24,AFF_1:39,41;
        then a1,b1 // a3,b3 by ANALMETR:36;
        then
A79:    x,y // a3,b3 by A32,A59,ANALMETR:60;
        M9 // N9 by A57,ANALMETR:46;
        then a19,b19 // a49,b49 by A5,A7,A10,A12,AFF_1:39;
        then a1,b1 // a4,b4 by ANALMETR:36;
        then
A80:    x,y // a4,b4 by A32,A59,ANALMETR:60;
        M9 // N9 by A57,ANALMETR:46;
        then a19,b19 // a29,b29 by A5,A7,A9,A11,AFF_1:39;
        then a1,b1 // a2,b2 by ANALMETR:36;
        then
A81:    a2,b2 // x,y by A32,A59,ANALMETR:60;
        a2,a3 // b2,b3 by A21,ANALMETR:59;
        then
A82:    x,a3 // y,b3 by A2,A77,A51,A75,A81,A58,A78;
        M9 // N9 by A57,ANALMETR:46;
        then a19,b19 // a49, b49 by A5,A7,A10,A12,AFF_1:39;
        then a1,b1 // a4,b4 by ANALMETR:36;
        then x,a4 // y,b4 by A2,A23,A59,A77,A55,A54,A64;
        hence thesis by A2,A77,A82,A65,A70,A79,A80;
      end;
      now
        assume not M // N;
        then not M9 // N9 by ANALMETR:46;
        then consider o9 being Element of the AffinStruct of X such that
A83:    o9 in M9 and
A84:    o9 in N9 by A24,A25,AFF_1:58;
        reconsider o=o9 as Element of X;
        consider K such that
A85:    o in K and
A86:    N _|_ K by A4,Th3;
        reconsider K9=K as Subset of the AffinStruct of X;
A87:    K is being_line by A86,ANALMETR:44;
        then
A88:    K9 is being_line by ANALMETR:43;
        now
A89:      not a2 in K & not a4 in K & not b2 in K & not b4 in K
          proof
A90:        now
              let x be Element of X;
              assume that
A91:          x in N and
A92:          o<>x;
              assume x in K;
              then o,x _|_ o,x by A84,A85,A86,A91,ANALMETR:56;
              hence contradiction by A92,ANALMETR:39;
            end;
            assume not thesis;
            hence contradiction by A9,A10,A11,A12,A13,A14,A15,A16,A83,A90;
          end;
A93:      LIN o,a2,b2 by A4,A9,A11,A84,Th4;
A94:      now
            assume LIN a1,b1,a4;
            then LIN a19,b19,a49 by ANALMETR:40;
            hence contradiction by A5,A7,A13,A24,A32,AFF_1:25;
          end;
A95:      LIN o,a3,b3 by A3,A6,A8,A83,Th4;
A96:      now
            assume LIN a3,b3,a2;
            then LIN a39,b39,a29 by ANALMETR:40;
            hence contradiction by A6,A8,A14,A24,A26,A32,AFF_1:25;
          end;
A97:      LIN o,a1,b1 by A3,A5,A7,A83,Th4;
A98:      now
            let x9 be Element of the AffinStruct of X;
            consider D9 being Subset of the AffinStruct of X such that
A99:        x9 in D9 and
A100:       N9 // D9 by A25,AFF_1:49;
            reconsider D=D9 as Subset of the carrier of X;
            N // D by A100,ANALMETR:46;
            then K _|_ D by A86,ANALMETR:52;
            then consider y being Element of X such that
A101:       y in K and
A102:       y in D by ANALMETR:66;
            reconsider y9=y as Element of the AffinStruct of X;
            take y9;
            thus y9 in K9 & x9,y9 // N9 by A99,A100,A101,A102,AFF_1:40;
          end;
          then consider d19 be Element of the AffinStruct of X such that
A103:     d19 in K9 and
A104:     a19,d19 // N9;
          consider e19 be Element of the AffinStruct of X such that
A105:     e19 in K9 and
A106:     b19,e19 // N9 by A98;
          consider d39 be Element of the AffinStruct of X such that
A107:     d39 in K9 and
A108:     a39,d39 // N9 by A98;
          consider e39 be Element of the AffinStruct of X such that
A109:     e39 in K9 and
A110:     b39,e39 // N9 by A98;
          reconsider d1=d19,d3=d39,e1=e19,e3=e39 as Element of X;
A111:     LIN o,d1,e1 by A85,A87,A103,A105,Th4;
A112:     o<>e1 & o<>e3 & o<>d1 & o<>d3
          proof
A113:       now
              let x,y be Element of X,
                  x9,y9 be Element of the AffinStruct of X;
              assume that
A114:         x9,y9 // N9 and
A115:         x=x9 and
A116:         y=y9 and
              x in M and
              y in K and
A117:         not x in N;
              assume o=y;
              then o9,x9 // N9 by A114,A116,AFF_1:34;
              hence contradiction by A25,A84,A115,A117,AFF_1:23;
            end;
            assume not thesis;
            hence
            contradiction by A5,A6,A7,A8,A17,A18,A19,A20,A103,A104,A107,A108
,A109,A110,A105,A106,A113;
          end;
          a1,d1 // o,a2 by A9,A84,A104,Th6;
          then
A118:     d1,a1 // o,a2 by ANALMETR:59;
A119:     not e1 in N by A19,A106,AFF_1:35;
A120:     not d3 in N by A18,A108,AFF_1:35;
          a19,d19 // b19,e19 by A25,A104,A106,AFF_1:31;
          then a1,d1 // b1,e1 by ANALMETR:36;
          then
A121:     d1,a1 // e1,b1 by ANALMETR:59;
          assume
A122:     K<>M;
A123:     now
            assume LIN a3,b3,d3;
            then LIN a39,b39,d39 by ANALMETR:40;
            then d3 in M by A6,A8,A24,A26,A32,AFF_1:25;
            hence contradiction by A3,A83,A85,A87,A122,A107,A112,Th5;
          end;
A124:     now
            assume LIN a1,b1,d1;
            then LIN a19,b19,d19 by ANALMETR:40;
            then d1 in M by A5,A7,A24,A32,AFF_1:25;
            hence contradiction by A3,A83,A85,A87,A122,A103,A112,Th5;
          end;
          a39,d39 // b39,e39 by A25,A108,A110,AFF_1:31;
          then
A125:     a3,d3 // b3,e3 by ANALMETR:36;
          then
A126:     d3,a3 // e3,b3 by ANALMETR:59;
A127:     not LIN d3,e3,a3 & not LIN d3,e3,a4
          proof
A128:       d3<>e3
            proof
              assume not thesis;
              then LIN e3,a3,b3 by A126,ANALMETR:def 10;
              then LIN e39,a39,b39 by ANALMETR:40;
              then LIN a39,b39,e39 by AFF_1:6;
              then e39 in M9 by A6,A8,A24,A26,A32,AFF_1:25;
              hence contradiction by A24,A83,A85,A88,A122,A109,A112,AFF_1:18;
            end;
            assume not thesis;
            then LIN d39,e39,a39 or LIN d39,e39,a49 by ANALMETR:40;
            then a39 in K9 or a49 in K9 by A88,A107,A109,A128,AFF_1:25;
            hence
            contradiction by A6,A10,A13,A18,A24,A25,A83,A84,A85,A86,A88,A122,
AFF_1:18;
          end;
A129:     now
            assume LIN a1,b1,a2;
            then LIN a19,b19,a29 by ANALMETR:40;
            hence contradiction by A5,A7,A14,A24,A32,AFF_1:25;
          end;
A130:     LIN o,a4,b4 by A4,A10,A12,A84,Th4;
          a1,a2 // b1,b2 by A22,ANALMETR:59;
          then d1,a2 // e1,b2 by A2,A14,A15,A17,A19,A83,A84,A112,A93,A97,A111
,A124,A129,A121,A118;
          then
A131:     a2,d1 // b2,e1 by ANALMETR:59;
A132:     not e3 in N by A20,A110,AFF_1:35;
A133:     LIN o,d3,e3 by A85,A87,A107,A109,Th4;
          a1,d1 // o,a4 by A10,A84,A104,Th6;
          then d1,a1 // o,a4 by ANALMETR:59;
          then
A134:     d1,a4 // e1,b4 by A2,A13,A16,A17,A19,A23,A83,A84,A112,A97,A130,A111
,A124,A94,A121;
A135:     a3,d3 // o,a4 by A10,A84,A108,Th6;
A136:     not d1 in N by A17,A104,AFF_1:35;
          a3,d3 // o,a2 by A9,A84,A108,Th6;
          then d3,a3 // o,a2 by ANALMETR:59;
          then d3,a2 // e3,b2 by A2,A14,A15,A18,A20,A21,A83,A84,A112,A93,A95
,A133,A123,A96,A126;
          then d3,a4 // e3,b4 by A1,A9,A10,A11,A12,A86,A103,A107,A109,A105,A131
,A134,A136,A120,A119,A132,A89;
          hence
          thesis by A2,A13,A16,A18,A20,A83,A84,A112,A130,A95,A133,A125,A127
,A135;
        end;
        hence
        thesis by A1,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,A18,A19,A20
,A21,A22,A23,A86;
      end;
      hence thesis by A40;
    end;
    now
A137: not LIN a19,b29,b19
      proof
        assume LIN a19,b29,b19;
        then LIN a19,b19,b29 by AFF_1:6;
        hence contradiction by A5,A7,A15,A24,A32,AFF_1:25;
      end;
      a19,b19 // a19,b39 by A5,A7,A8,A24,AFF_1:39,41;
      then
A138: LIN a19,b19,b39 by AFF_1:def 1;
      assume
A139: a1=a3;
      then a2,a1 // b2,b3 by A21,ANALMETR:59;
      then b2,b1 // b2,b3 by A5,A14,A22,ANALMETR:60;
      then b29,b19 // b29,b39 by ANALMETR:36;
      hence thesis by A23,A139,A137,A138,AFF_1:14;
    end;
    hence thesis by A33,A37;
  end;
A140: a1=b1 implies a2=b2 & a3=b3 & a4=b4
  proof
    assume
A141: a1=b1;
A142: now
      LIN a1,a4,b4 by A23,A141,ANALMETR:def 10;
      then LIN a19,a49,b49 by ANALMETR:40;
      then
A143: LIN a49,b49,a19 by AFF_1:6;
      assume a4<>b4;
      hence contradiction by A10,A12,A17,A25,A143,AFF_1:25;
    end;
A144: now
      a1,a2 // a1,b2 by A22,A141,ANALMETR:59;
      then LIN a1,a2,b2 by ANALMETR:def 10;
      then LIN a19,a29,b29 by ANALMETR:40;
      then
A145: LIN a29,b29,a19 by AFF_1:6;
      assume a2<>b2;
      hence contradiction by A9,A11,A17,A25,A145,AFF_1:25;
    end;
A146: now
      a2,a3 // a2,b3 by A21,A144,ANALMETR:59;
      then LIN a2,a3,b3 by ANALMETR:def 10;
      then LIN a29,a39,b39 by ANALMETR:40;
      then
A147: LIN a39,b39,a29 by AFF_1:6;
      assume a3<>b3;
      hence contradiction by A6,A8,A14,A24,A147,AFF_1:25;
    end;
    assume a2<>b2 or a3<>b3 or a4<>b4;
    hence contradiction by A144,A142,A146;
  end;
  now
    assume a1=b1;
    then a39,a49 // b39,b49 by A140,AFF_1:2;
    hence thesis by ANALMETR:36;
  end;
  hence thesis by A31;
end;
