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_AH implies X is satisfying_TDES
proof
  assume
A1: X is satisfying_AH;
  let o,a,a1,b,b1,c,c1;
  assume that
A2: o<>a and
A3: o<>a1 and
A4: o<>b and
A5: o<>b1 and
A6: o<>c and
A7: o<>c1 and
A8: not LIN b,b1,a and
A9: not LIN b,b1,c and
A10: LIN o,a,a1 and
A11: LIN o,b,b1 and
A12: LIN o,c,c1 and
A13: a,b // a1,b1 and
A14: a,b // o,c and
A15: b,c // b1,c1;
  consider e1 such that
A16: o,b _|_ o,e1 and
A17: o<>e1 by ANALMETR:def 9;
  consider c2 such that
A18: o,c _|_ o,c2 and
A19: o<>c2 by ANALMETR:def 9;
  consider e2 such that
A20: b,c _|_ c2,e2 and
A21: c2<>e2 by ANALMETR:def 9;
  reconsider o9=o,a9=a,a19=a1,b9=b,b19=b1,c9 = c,c19= c1,c29= c2,e19=e1, e29=
  e2 as Element of the AffinStruct of X;
A22: b1<>a1
  proof
    assume b1=a1;
    then LIN o9,b9,a19 by A11,ANALMETR:40;
    then o9,b9 // o9,a19 by AFF_1:def 1;
    then o9,a19 // o9,b9 by AFF_1:4;
    then
A23: o,a1 // o,b by ANALMETR:36;
    o,a // o,a1 by A10,ANALMETR:def 10;
    then o9,a9 // o9,a19 by ANALMETR:36;
    then o9,a19 // o9,a9 by AFF_1:4;
    then o,a1 // o,a by ANALMETR:36;
    then o,a // o,b by A3,A23,ANALMETR:60;
    then LIN o,a,b by ANALMETR:def 10;
    then
A24: LIN o9,a9,b9 by ANALMETR:40;
    then LIN b9,o9,a9 by AFF_1:6;
    then b9,o9 // b9,a9 by AFF_1:def 1;
    then o9,b9 // a9,b9 by AFF_1:4;
    then
A25: o,b // a,b by ANALMETR:36;
A26: a9<>b9
    proof
      assume a9=b9;
      then LIN b9,b19,a9 by AFF_1:7;
      hence contradiction by A8,ANALMETR:40;
    end;
    o,b // o,b1 by A11,ANALMETR:def 10;
    then a,b // o,b1 by A4,A25,ANALMETR:60;
    then
A27: a9,b9 // o9,b19 by ANALMETR:36;
    LIN a9,b9,o9 by A24,AFF_1:6;
    then LIN a9,b9,b19 by A27,A26,AFF_1:9;
    then LIN b9,b19,a9 by AFF_1:6;
    hence contradiction by A8,ANALMETR:40;
  end;
A28: now
    not o9,e19 // c29,e29
    proof
      assume o9,e19 // c29,e29;
      then o,e1 // c2,e2 by ANALMETR:36;
      then c2,e2 _|_ o,b by A16,A17,ANALMETR:62;
      then c2,e2 _|_ b,o by ANALMETR:61;
      then b,c // b,o by A20,A21,ANALMETR:63;
      then LIN b,c,o by ANALMETR:def 10;
      then LIN b9,c9,o9 by ANALMETR:40;
      then LIN b9,o9,c9 by AFF_1:6;
      then LIN b,o,c by ANALMETR:40;
      then
A29:  b,o // b,c by ANALMETR:def 10;
      LIN o9,b9,b19 by A11,ANALMETR:40;
      then LIN b9,o9,b19 by AFF_1:6;
      then LIN b,o,b1 by ANALMETR:40;
      then b,o // b,b1 by ANALMETR:def 10;
      then b,b1 // b,c by A4,A29,ANALMETR:60;
      hence contradiction by A9,ANALMETR:def 10;
    end;
    then consider b29 such that
A30: LIN o9,e19,b29 and
A31: LIN c29,e29,b29 by AFF_1:60;
    reconsider b2=b29 as Element of X;
    LIN o,e1,b2 by A30,ANALMETR:40;
    then o,e1 // o,b2 by ANALMETR:def 10;
    then
A32: o,b _|_ o,b2 by A16,A17,ANALMETR:62;
    o,b // o,b1 by A11,ANALMETR:def 10;
    then
A33: o,b1 _|_ o,b2 by A4,A32,ANALMETR:62;
    assume
A34: not LIN a,b,c;
A35: b<>a
    proof
      assume b=a;
      then LIN a9,b9,c9 by AFF_1:7;
      hence contradiction by A34,ANALMETR:40;
    end;
A36: not o,c // o,b
    proof
      assume o,c // o,b;
      then LIN o,c,b by ANALMETR:def 10;
      then LIN o9,c9,b9 by ANALMETR:40;
      then LIN c9,o9,b9 by AFF_1:6;
      then c9,o9 // c9,b9 by AFF_1:def 1;
      then o9,c9 // b9,c9 by AFF_1:4;
      then
A37:  o,c // b,c by ANALMETR:36;
      a9,b9 // o9,c9 by A14,ANALMETR:36;
      then o9,c9 // b9,a9 by AFF_1:4;
      then o,c // b,a by ANALMETR:36;
      then b,a // b,c by A6,A37,ANALMETR:60;
      then LIN b,a,c by ANALMETR:def 10;
      then LIN b9,a9,c9 by ANALMETR:40;
      then LIN a9,b9,c9 by AFF_1:6;
      hence contradiction by A34,ANALMETR:40;
    end;
A38: not o,a // o,c
    proof
      assume o,a // o,c;
      then LIN o,a,c by ANALMETR:def 10;
      then LIN o9,a9,c9 by ANALMETR:40;
      then LIN c9,o9,a9 by AFF_1:6;
      then LIN c,o,a by ANALMETR:40;
      then
A39:  c,o // c,a by ANALMETR:def 10;
      a9,b9 // o9,c9 by A14,ANALMETR:36;
      then c9,o9 // a9,b9 by AFF_1:4;
      then c,o // a,b by ANALMETR:36;
      then a,b // c,a by A6,A39,ANALMETR:60;
      then a9,b9 // c9,a9 by ANALMETR:36;
      then a9,b9 // a9,c9 by AFF_1:4;
      then a,b // a,c by ANALMETR:36;
      hence contradiction by A34,ANALMETR:def 10;
    end;
    LIN c2,e2,b2 by A31,ANALMETR:40;
    then c2,e2 // c2,b2 by ANALMETR:def 10;
    then
A40: b,c _|_ c2,b2 by A20,A21,ANALMETR:62;
    consider e1 such that
A41: o,a _|_ o,e1 and
A42: o<>e1 by ANALMETR:def 9;
    consider e2 such that
A43: a,c _|_ c2,e2 and
A44: c2<>e2 by ANALMETR:def 9;
    reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
    not o9,e19 // c29,e29
    proof
      assume o9,e19 // c29,e29;
      then o,e1 // c2,e2 by ANALMETR:36;
      then c2,e2 _|_ o,a by A41,A42,ANALMETR:62;
      then a,c // o,a by A43,A44,ANALMETR:63;
      then a9,c9 // o9,a9 by ANALMETR:36;
      then a9,c9 // a9,o9 by AFF_1:4;
      then LIN a9,c9,o9 by AFF_1:def 1;
      then LIN c9,o9,a9 by AFF_1:6;
      then LIN c,o,a by ANALMETR:40;
      then
A45:  c,o // c,a by ANALMETR:def 10;
      a9,b9 // o9,c9 by A14,ANALMETR:36;
      then c9,o9 // a9,b9 by AFF_1:4;
      then c,o // a,b by ANALMETR:36;
      then a,b // c,a by A6,A45,ANALMETR:60;
      then a9,b9 // c9,a9 by ANALMETR:36;
      then a9,b9 // a9,c9 by AFF_1:4;
      then a,b // a,c by ANALMETR:36;
      hence contradiction by A34,ANALMETR:def 10;
    end;
    then consider a29 such that
A46: LIN o9,e19,a29 and
A47: LIN c29,e29,a29 by AFF_1:60;
    reconsider a2=a29 as Element of X;
    LIN o,e1,a2 by A46,ANALMETR:40;
    then o,e1 // o,a2 by ANALMETR:def 10;
    then
A48: o,a _|_ o,a2 by A41,A42,ANALMETR:62;
A49: c2<>a2
    proof
      assume c2=a2;
      then o,c // o,a by A18,A19,A48,ANALMETR:63;
      then LIN o,c,a by ANALMETR:def 10;
      then LIN o9,c9,a9 by ANALMETR:40;
      then LIN c9,a9,o9 by AFF_1:6;
      then c9,a9 // c9,o9 by AFF_1:def 1;
      then o9,c9 // a9,c9 by AFF_1:4;
      then o,c // a,c by ANALMETR:36;
      then a,b // a,c by A6,A14,ANALMETR:60;
      hence contradiction by A34,ANALMETR:def 10;
    end;
    LIN c2,e2,a2 by A47,ANALMETR:40;
    then c2,e2 // c2,a2 by ANALMETR:def 10;
    then a,c _|_ c2,a2 by A43,A44,ANALMETR:62;
    then
A50: c,a _|_ c2,a2 by ANALMETR:61;
A51: not LIN o9,b29,a29
    proof
A52:  o<>b2
      proof
        a9,b9 // o9,c9 by A14,ANALMETR:36;
        then o9,c9 // b9,a9 by AFF_1:4;
        then
A53:    o,c // b,a by ANALMETR:36;
        assume o=b2;
        then o,c2 _|_ b,c by A40,ANALMETR:61;
        then o,c // b,c by A18,A19,ANALMETR:63;
        then b,a // b,c by A6,A53,ANALMETR:60;
        then LIN b,a,c by ANALMETR:def 10;
        then LIN b9,a9,c9 by ANALMETR:40;
        then LIN a9,b9,c9 by AFF_1:6;
        hence contradiction by A34,ANALMETR:40;
      end;
A54:  o<>a2
      proof
        assume
A55:    o=a2;
        c2,o _|_ o,c by A18,ANALMETR:61;
        then o,c // c,a by A19,A50,A55,ANALMETR:63;
        then a,b // c,a by A6,A14,ANALMETR:60;
        then a9,b9 // c9,a9 by ANALMETR:36;
        then a9,b9 // a9,c9 by AFF_1:4;
        then a,b // a,c by ANALMETR:36;
        hence contradiction by A34,ANALMETR:def 10;
      end;
      assume LIN o9,b29,a29;
      then LIN o,b2,a2 by ANALMETR:40;
      then o,b2 // o,a2 by ANALMETR:def 10;
      then o,a2 _|_ o,b by A32,A52,ANALMETR:62;
      then o,a // o,b by A48,A54,ANALMETR:63;
      then LIN o,a,b by ANALMETR:def 10;
      then LIN o9,a9,b9 by ANALMETR:40;
      then
A56:  LIN a9,b9,o9 by AFF_1:6;
A57:  a9<>b9
      proof
        assume a9=b9;
        then LIN a9,b9,c9 by AFF_1:7;
        hence contradiction by A34,ANALMETR:40;
      end;
      a9,b9 // o9,c9 by A14,ANALMETR:36;
      then LIN a9,b9,c9 by A56,A57,AFF_1:9;
      hence contradiction by A34,ANALMETR:40;
    end;
    consider e1 such that
A58: o,a1 _|_ o,e1 and
A59: o<>e1 by ANALMETR:def 9;
    consider e2 such that
A60: a1,c1 _|_ c2,e2 and
A61: c2<>e2 by ANALMETR:def 9;
    reconsider e19=e1,e29=e2 as Element of the AffinStruct of X;
    not o9,e19 // c29,e29
    proof
      assume o9,e19 // c29,e29;
      then o,e1 // c2,e2 by ANALMETR:36;
      then c2,e2 _|_ o,a1 by A58,A59,ANALMETR:62;
      then a1,c1 // o,a1 by A60,A61,ANALMETR:63;
      then a19,c19 // o9,a19 by ANALMETR:36;
      then a19,c19 // a19,o9 by AFF_1:4;
      then LIN a19,c19,o9 by AFF_1:def 1;
      then LIN o9,c19,a19 by AFF_1:6;
      then o9,c19 // o9,a19 by AFF_1:def 1;
      then
A62:  o,c1 // o,a1 by ANALMETR:36;
      LIN o9,a9,a19 by A10,ANALMETR:40;
      then o9,a9 // o9,a19 by AFF_1:def 1;
      then o9,a19 // o9,a9 by AFF_1:4;
      then
A63:  o,a1 // o,a by ANALMETR:36;
      LIN o9,c9,c19 by A12,ANALMETR:40;
      then o9,c9 // o9,c19 by AFF_1:def 1;
      then o9,c19 // o9,c9 by AFF_1:4;
      then o,c1 // o,c by ANALMETR:36;
      then o,a1 // o,c by A7,A62,ANALMETR:60;
      then o,a // o,c by A3,A63,ANALMETR:60;
      then LIN o,a,c by ANALMETR:def 10;
      then LIN o9,a9,c9 by ANALMETR:40;
      then LIN c9,o9,a9 by AFF_1:6;
      then c9,o9 // c9,a9 by AFF_1:def 1;
      then o9,c9 // a9,c9 by AFF_1:4;
      then o,c // a,c by ANALMETR:36;
      then a,b // a,c by A6,A14,ANALMETR:60;
      hence contradiction by A34,ANALMETR:def 10;
    end;
    then consider a39 such that
A64: LIN o9,e19,a39 and
A65: LIN c29,e29,a39 by AFF_1:60;
    reconsider a3=a39 as Element of X;
    LIN o,e1,a3 by A64,ANALMETR:40;
    then o,e1 // o,a3 by ANALMETR:def 10;
    then
A66: o,a1 _|_ o,a3 by A58,A59,ANALMETR:62;
    b<>c
    proof
      assume b= c;
      then LIN a9,b9,c9 by AFF_1:7;
      hence contradiction by A34,ANALMETR:40;
    end;
    then c2,b2 _|_ b1,c1 by A15,A40,ANALMETR:62;
    then
A67: c1,b1 _|_ c2,b2 by ANALMETR:61;
A68: not o,a1 // o,c1
    proof
      o,a // o,a1 by A10,ANALMETR:def 10;
      then o9,a9 // o9,a19 by ANALMETR:36;
      then o9,a19 // o9,a9 by AFF_1:4;
      then
A69:  o,a1 // o,a by ANALMETR:36;
      assume o,a1 // o,c1;
      then o9,a19 // o9,c19 by ANALMETR:36;
      then o9,c19 // o9,a19 by AFF_1:4;
      then
A70:  o,c1 // o,a1 by ANALMETR:36;
      o,c // o,c1 by A12,ANALMETR:def 10;
      then o9,c9 // o9,c19 by ANALMETR:36;
      then o9,c19 // o9,c9 by AFF_1:4;
      then o,c1 // o,c by ANALMETR:36;
      then o,a1 // o,c by A7,A70,ANALMETR:60;
      then o,a // o,c by A3,A69,ANALMETR:60;
      then LIN o,a,c by ANALMETR:def 10;
      then LIN o9,a9,c9 by ANALMETR:40;
      then LIN c9,a9,o9 by AFF_1:6;
      then c9,a9 // c9,o9 by AFF_1:def 1;
      then o9,c9// a9,c9 by AFF_1:4;
      then o,c // a,c by ANALMETR:36;
      then a,b // a,c by A6,A14,ANALMETR:60;
      hence contradiction by A34,ANALMETR:def 10;
    end;
    a9,b9 // o9,c9 by A14,ANALMETR:36;
    then o9,c9 // b9,a9 by AFF_1:4;
    then
A71: o,c // b,a by ANALMETR:36;
A72: not o,c1 // o,b1
    proof
      o,b // o,b1 by A11,ANALMETR:def 10;
      then o9,b9 // o9,b19 by ANALMETR:36;
      then o9,b19 // o9,b9 by AFF_1:4;
      then
A73:  o,b1 // o,b by ANALMETR:36;
      o,c // o,c1 by A12,ANALMETR:def 10;
      then o9,c9 // o9,c19 by ANALMETR:36;
      then o9,c19 // o9,c9 by AFF_1:4;
      then
A74:  o,c1 // o,c by ANALMETR:36;
      assume o,c1 // o,b1;
      then o,b1 // o,c by A7,A74,ANALMETR:60;
      then o,b // o,c by A5,A73,ANALMETR:60;
      then LIN o,b,c by ANALMETR:def 10;
      then LIN o9,b9,c9 by ANALMETR:40;
      then LIN c9,o9,b9 by AFF_1:6;
      then c9,o9 // c9,b9 by AFF_1:def 1;
      then o9,c9 // b9,c9 by AFF_1:4;
      then
A75:  o,c // b,c by ANALMETR:36;
      a9,b9 // o9,c9 by A14,ANALMETR:36;
      then o9,c9 // b9,a9 by AFF_1:4;
      then o,c // b,a by ANALMETR:36;
      then b,a // b,c by A6,A75,ANALMETR:60;
      then LIN b,a,c by ANALMETR:def 10;
      then LIN b9,a9,c9 by ANALMETR:40;
      then LIN a9,b9,c9 by AFF_1:6;
      hence contradiction by A34,ANALMETR:40;
    end;
    a<>b
    proof
      assume a=b;
      then LIN b9,b19,a9 by AFF_1:7;
      hence contradiction by A8,ANALMETR:40;
    end;
    then o,c // a1,b1 by A13,A14,ANALMETR:60;
    then o9,c9 // a19,b19 by ANALMETR:36;
    then o9,c9 // b19,a19 by AFF_1:4;
    then
A76: o,c // b1,a1 by ANALMETR:36;
    o,c // o,c1 by A12,ANALMETR:def 10;
    then
A77: o,c1 // b1,a1 by A6,A76,ANALMETR:60;
    o,c // o,c1 by A12,ANALMETR:def 10;
    then
A78: o,c1 _|_ o,c2 by A6,A18,ANALMETR:62;
    c,b _|_ c2,b2 by A40,ANALMETR:61;
    then b,a _|_ b2,a2 by A1,A18,A32,A48,A71,A50,A38,A36,CONAFFM:def 2;
    then b2,a2 _|_ a,b by ANALMETR:61;
    then b2,a2 _|_ a1,b1 by A13,A35,ANALMETR:62;
    then
A79: b1,a1 _|_ b2,a2 by ANALMETR:61;
    o,a // o,a1 by A10,ANALMETR:def 10;
    then o,a1 _|_ o,a2 by A2,A48,ANALMETR:62;
    then o,a2 // o,a3 by A3,A66,ANALMETR:63;
    then LIN o,a2,a3 by ANALMETR:def 10;
    then
A80: LIN o9,a29,a39 by ANALMETR:40;
    LIN c2,e2,a3 by A65,ANALMETR:40;
    then c2,e2 // c2,a3 by ANALMETR:def 10;
    then
A81: a1,c1 _|_ c2,a3 by A60,A61,ANALMETR:62;
    then c1,a1 _|_ c2,a3 by ANALMETR:61;
    then b1,a1 _|_ b2,a3 by A1,A66,A78,A33,A67,A77,A68,A72,CONAFFM:def 2;
    then b2,a3 // b2,a2 by A22,A79,ANALMETR:63;
    then b29,a39 // b29,a29 by ANALMETR:36;
    then b29,a29 // b29,a39 by AFF_1:4;
    then a29=a39 by A51,A80,AFF_1:14;
    then c,a // a1,c1 by A50,A81,A49,ANALMETR:63;
    then c9,a9 // a19,c19 by ANALMETR:36;
    then a9,c9 // a19,c19 by AFF_1:4;
    hence thesis by ANALMETR:36;
  end;
  now
    assume
A82: LIN a,b,c;
    then
A83: LIN a9,b9,c9 by ANALMETR:40;
A84: a<>b
    proof
      assume a=b;
      then LIN b9,b19,a9 by AFF_1:7;
      hence contradiction by A8,ANALMETR:40;
    end;
A85: now
      assume a1,c1 // a,b;
      then a19,c19 // a9,b9 by ANALMETR:36;
      then a9,b9 // a19,c19 by AFF_1:4;
      then
A86:  a,b // a1,c1 by ANALMETR:36;
      a,b // a,c by A82,ANALMETR:def 10;
      hence thesis by A84,A86,ANALMETR:60;
    end;
A87: b<>c
    proof
      assume b=c;
      then LIN b9,b19,c9 by AFF_1:7;
      hence contradiction by A9,ANALMETR:40;
    end;
A88: now
      LIN c9,b9,a9 by A83,AFF_1:6;
      then c9,b9 // c9,a9 by AFF_1:def 1;
      then b9,c9 // a9,c9 by AFF_1:4;
      then
A89:  b,c // a,c by ANALMETR:36;
      assume a1=b1;
      hence thesis by A15,A87,A89,ANALMETR:60;
    end;
    LIN b9,a9,c9 by A83,AFF_1:6;
    then b9,a9 // b9,c9 by AFF_1:def 1;
    then a9,b9 // b9,c9 by AFF_1:4;
    then a,b // b,c by ANALMETR:36;
    then b,c // a1,b1 by A13,A84,ANALMETR:60;
    then b1,c1 // a1,b1 by A15,A87,ANALMETR:60;
    then b19,c19 // a19,b19 by ANALMETR:36;
    then b19,a19 // b19,c19 by AFF_1:4;
    then LIN b19,a19,c19 by AFF_1:def 1;
    then LIN a19,b19,c19 by AFF_1:6;
    then LIN a1,b1,c1 by ANALMETR:40;
    then
A90: a1,b1 // a1,c1 by ANALMETR:def 10;
    a9,b9 // a19,b19 by A13,ANALMETR:36;
    then a19,b19 // a9,b9 by AFF_1:4;
    then a1,b1 // a,b by ANALMETR:36;
    hence thesis by A90,A85,A88,ANALMETR:60;
  end;
  hence thesis by A28;
end;
