reserve X for AffinPlane;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,d,d1,d2, d3,d4,d5,e1,e2,x,y,z
  for Element of X;
reserve Y,Z,M,N,A,K,C for Subset of X;
reserve X for OrtAfPl;
reserve o9,a9,a19,a29,a39,a49,b9,b19,b29,b39,b49,c9,c19 for Element of X;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1 for Element of the AffinStruct of X;
reserve M9,N9 for Subset of X;
reserve A,M,N for Subset of the AffinStruct of X;

theorem
  X is satisfying_TDES iff the AffinStruct of X is Moufangian
proof
  the AffinStruct of X is Moufangian implies X is satisfying_TDES
  proof
    assume
A1: the AffinStruct of X is Moufangian;
    now
      let o9,a9,a19,b9,b19,c9,c19;
      assume that
      o9<>a9 and
      o9<>a19 and
A2:   o9<>b9 and
      o9<>b19 and
A3:   o9<>c9 and
      o9<>c19 and
A4:   not LIN b9,b19,a9 and
A5:   not LIN b9,b19,c9 and
A6:   LIN o9,a9,a19 and
A7:   LIN o9,b9,b19 and
A8:   LIN o9,c9,c19 and
A9:   a9,b9 // a19,b19 and
A10:  a9,b9 // o9,c9 and
A11:  b9,c9 // b19,c19;
      reconsider o=o9,a=a9,a1=a19,b=b9,b1=b19,c =c9,c1=c19 as Element
 of the AffinStruct of X;
A12:  LIN o,b,b1 by A7,ANALMETR:40;
      LIN o,c,c1 by A8,ANALMETR:40;
      then consider M such that
A13:  M is being_line and
A14:  o in M and
A15:  c in M and
A16:  c1 in M by AFF_1:21;
A17:  not LIN b,b1,c by A5,ANALMETR:40;
A18:  not b in M
      proof
        LIN b,b1,o by A12,AFF_1:6;
        then b,b1 // b,o by AFF_1:def 1;
        then
A19:    b,b1 // o,b by AFF_1:4;
        assume b in M;
        then o,b // b,c by A13,A14,A15,AFF_1:39,41;
        then b,b1 // b,c by A2,A19,AFF_1:5;
        hence contradiction by A17,AFF_1:def 1;
      end;
      a,b // a1,b1 by A9,ANALMETR:36;
      then
A20:  b,a // b1,a1 by AFF_1:4;
      not LIN b,b1,a by A4,ANALMETR:40;
      then
A21:  b<>a by AFF_1:7;
A22:  b,c // b1,c1 by A11,ANALMETR:36;
A23:  LIN o,a,a1 by A6,ANALMETR:40;
      a,b // o,c by A10,ANALMETR:36;
      then b,a // o,c by AFF_1:4;
      then b,a // M by A3,A13,A14,A15,AFF_1:27;
      then a,c // a1,c1 by A1,A3,A23,A12,A13,A14,A15,A16,A18,A21,A20,A22,
AFF_2:def 7;
      hence a9,c9 // a19,c19 by ANALMETR:36;
    end;
    hence thesis by CONMETR:def 5;
  end;
  hence thesis by CONMETR:7;
end;
