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 Th7:
  X is satisfying_TDES implies the AffinStruct of X is Moufangian
proof
  assume
A1: X is satisfying_TDES;
  let K9 be Subset of the AffinStruct of X;
  let o9,a9,b9,c9,a19,b19,c19 be Element of the AffinStruct of X;
  assume that
A2: K9 is being_line and
A3: o9 in K9 and
A4: c9 in K9 and
A5: c19 in K9 and
A6: not a9 in K9 and
A7: o9<>c9 and
A8: a9<>b9 and
A9: LIN o9,a9,a19 and
A10: LIN o9,b9,b19 and
A11: a9,b9 // a19,b19 and
A12: a9,c9 // a19,c19 and
A13: a9,b9 // K9;
  reconsider K=K9 as Subset of X;
  reconsider o=o9,a=a9,a1=a19,b=b9,b1=b19,c =c9,c1=c19 as Element
 of X;
A14: a,c // a1,c1 by A12,ANALMETR:36;
A15: a,b // a1,b1 by A11,ANALMETR:36;
  now
A16: now
      assume
A17:  o<>a1;
A18:  o<>a & o<>b & o<>c & o<>c1 & o<>b1
      proof
A19:    now
          o9,c9 // K9 by A2,A3,A4,AFF_1:23;
          then
A20:      a9,b9 // o9,c9 by A2,A13,AFF_1:31;
          assume o=b1;
          then a19,o9 // o9,c9 by A8,A11,A20,DIRAF:40;
          then o9,c9 // o9,a19 by AFF_1:4;
          then LIN o9,c9,a19 by AFF_1:def 1;
          then
A21:      a19 in K9 by A2,A3,A4,A7,AFF_1:25;
          LIN o9,a19,a9 by A9,AFF_1:6;
          hence contradiction by A2,A3,A6,A17,A21,AFF_1:25;
        end;
A22:    now
          o9,a9 // o9,a19 by A9,AFF_1:def 1;
          then
A23:      o9,a19 // a9, o9 by AFF_1:4;
          assume o=c1;
          then o9,a19 // a9,c9 by A12,AFF_1:4;
          then a9,c9 // a9,o9 by A17,A23,DIRAF:40;
          then LIN a9,c9,o9 by AFF_1:def 1;
          then LIN o9,c9,a9 by AFF_1:6;
          hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
        end;
        assume o=a or o=b or o=c or o=c1 or o=b1;
        hence contradiction by A3,A6,A7,A13,A22,A19,AFF_1:35;
      end;
A24:  now
        assume
A25:    a<>a19;
A26:    not LIN a,a1,b & not LIN a,a1,c
        proof
A27:      now
            LIN a9,a19,o9 by A9,AFF_1:6;
            then a9,a19 // a9,o9 by AFF_1:def 1;
            then
A28:        a,a1 // a,o by ANALMETR:36;
            assume LIN a,a1,b;
            then a,a1 // a,b by ANALMETR:def 10;
            then a,b // a,o by A25,A28,ANALMETR:60;
            then
A29:        a,b // o,a by ANALMETR:59;
            a9,b9 // o9,c9 by A2,A3,A4,A7,A13,AFF_1:27;
            then a,b // o,c by ANALMETR:36;
            then o,c // o,a by A8,A29,ANALMETR:60;
            then LIN o,c,a by ANALMETR:def 10;
            then LIN o9,c9,a9 by ANALMETR:40;
            hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
          end;
A30:      now
            LIN a9,a19,o9 by A9,AFF_1:6;
            then a9,a19 // a9,o9 by AFF_1:def 1;
            then
A31:        a,a1 // a,o by ANALMETR:36;
            assume LIN a,a1,c;
            then a,a1 // a,c by ANALMETR:def 10;
            then a,c // a,o by A25,A31,ANALMETR:60;
            then LIN a,c,o by ANALMETR:def 10;
            then LIN a9,c9,o9 by ANALMETR:40;
            then LIN c9,o9,a9 by AFF_1:6;
            hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
          end;
          assume LIN a,a1,b or LIN a,a1,c;
          hence contradiction by A27,A30;
        end;
A32:    LIN o,b,b1 by A10,ANALMETR:40;
        o9,c9 // o9,c19 by A2,A3,A4,A5,AFF_1:39,41;
        then o,c // o,c1 by ANALMETR:36;
        then
A33:    LIN o,c,c1 by ANALMETR:def 10;
        o9,c9 // K9 by A2,A3,A4,AFF_1:40,41;
        then a9,b9 // o9,c9 by A2,A13,AFF_1:31;
        then b9,a9 // o9,c9 by AFF_1:4;
        then
A34:    b,a // o,c by ANALMETR:36;
A35:    b,a // b1,a1 by A15,ANALMETR:59;
        LIN o,a,a1 by A9,ANALMETR:40;
        then b,c // b1,c1 by A1,A14,A17,A18,A26,A32,A33,A35,A34;
        hence thesis by ANALMETR:36;
      end;
      now
A36:    not LIN o9,a9,c9
        proof
          assume LIN o9,a9,c9;
          then LIN o9,c9,a9 by AFF_1:6;
          hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
        end;
        assume
A37:    a=a1;
        o9,c9 // o9,c19 by A2,A3,A4,A5,AFF_1:39,41;
        then LIN o9,c9,c19 by AFF_1:def 1;
        then
A38:    c9=c19 by A12,A37,A36,AFF_1:14;
        not LIN o9,a9,b9
        proof
          assume LIN o9,a9,b9;
          then LIN a9,b9,o9 by AFF_1:6;
          then
A39:      a9,b9 // a9,o9 by AFF_1:def 1;
          o9,c9 // K9 by A2,A3,A4,AFF_1:23;
          then a9,b9 // o9,c9 by A2,A13,AFF_1:31;
          then a9,o9 // o9,c9 by A8,A39,DIRAF:40;
          then o9,c9 // o9,a9 by AFF_1:4;
          then LIN o9,c9,a9 by AFF_1:def 1;
          hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
        end;
        then b9=b19 by A10,A11,A37,AFF_1:14;
        hence thesis by A38,AFF_1:2;
      end;
      hence thesis by A24;
    end;
    assume
A40: not b in K;
A41: o=a1 implies o=b1 & o=c1
    proof
      assume that
A42:  o=a1 and
A43:  o<>b1 or o<>c1;
A44:  now
A45:    o9,c19 // a9,c9 by A12,A42,AFF_1:4;
        assume
A46:    o<>c1;
        o9,c19 // o9,c9 by A2,A3,A4,A5,AFF_1:39,41;
        then a9,c9 // o9,c9 by A46,A45,DIRAF:40;
        then c9,a9 // c9,o9 by AFF_1:4;
        then LIN c9,a9,o9 by AFF_1:def 1;
        then LIN o9,c9,a9 by AFF_1:6;
        hence contradiction by A2,A3,A4,A6,A7,AFF_1:25;
      end;
      now
        o9,c9 // K9 by A2,A3,A4,AFF_1:23;
        then a9,b9 // o9,c9 by A2,A13,AFF_1:31;
        then o9,c9 // o9,b19 by A8,A11,A42,DIRAF:40;
        then LIN o9,c9,b19 by AFF_1:def 1;
        then
A47:    b19 in K9 by A2,A3,A4,A7,AFF_1:25;
        assume
A48:    o<>b1;
        LIN o9,b19,b9 by A10,AFF_1:6;
        hence contradiction by A2,A3,A40,A48,A47,AFF_1:25;
      end;
      hence contradiction by A43,A44;
    end;
    now
      assume o=a1;
      then b,c // b1,c1 by A41,ANALMETR:39;
      hence thesis by ANALMETR:36;
    end;
    hence thesis by A16;
  end;
  hence thesis by A6,A13,AFF_1:35;
end;
