reserve AP for AffinPlane,
  a,a9,b,b9,c,c9,x,y,o,p,q,r,s for Element of AP,
  A,C,C9,D,K,M,N,P,T for Subset of AP;

theorem Th3:
  AP is Moufangian implies AP is satisfying_TDES_1
proof
  assume
A1: AP is Moufangian;
    let K,o,a,b,c,a9,b9,c9;
    assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: c9 in K and
A6: not a in K and
A7: o<>c and
A8: a<>b and
A9: LIN o,a,a9 and
A10: a,b // a9,b9 and
A11: b,c // b9,c9 and
A12: a,c // a9,c9 and
A13: a,b // K;
    consider P such that
A14: a9 in P and
A15: K // P by A2,AFF_1:49;
A16: P is being_line by A15,AFF_1:36;
    set A=Line(o,b), C=Line(o,a);
A17: o in A & b in A by AFF_1:15;
    assume
A18: not LIN o,b,b9;
    then o<>b by AFF_1:7;
    then
A19: A is being_line by AFF_1:def 3;
A20: not b in K by A6,A13,AFF_1:35;
    not A // P
    proof
      assume A // P;
      then A // K by A15,AFF_1:44;
      hence contradiction by A3,A20,A17,AFF_1:45;
    end;
    then consider x such that
A21: x in A and
A22: x in P by A19,A16,AFF_1:58;
A23: o in C & a in C by AFF_1:15;
A24: LIN o,b,x by A19,A17,A21,AFF_1:21;
    a,b // P by A13,A15,AFF_1:43;
    then a9,b9 // P by A8,A10,AFF_1:32;
    then
A25: b9 in P by A14,A16,AFF_1:23;
    then
A26: LIN b9,x,b9 by A16,A22,AFF_1:21;
A27: C is being_line by A3,A6,AFF_1:def 3;
    then
A28: a9 in C by A3,A6,A9,A23,AFF_1:25;
A29: b9<>c9
    proof
A30:  a9,c9 // c,a by A12,AFF_1:4;
      assume
A31:  b9=c9;
      then P=K by A5,A15,A25,AFF_1:45;
      then a9=o by A2,A3,A6,A27,A23,A28,A14,AFF_1:18;
      then b9=o by A2,A3,A4,A5,A6,A31,A30,AFF_1:48;
      hence contradiction by A18,AFF_1:7;
    end;
A32: b<>c by A4,A6,A13,AFF_1:35;
    a9,x // K by A14,A15,A22,AFF_1:40;
    then a,b // a9,x by A2,A13,AFF_1:31;
    then b,c // x,c9 by A1,A2,A3,A4,A5,A6,A7,A8,A9,A12,A13,A24;
    then b9,c9 // x,c9 by A11,A32,AFF_1:5;
    then c9,b9 // c9,x by AFF_1:4;
    then LIN c9,b9,x by AFF_1:def 1;
    then
A33: LIN b9,x,c9 by AFF_1:6;
A34: a9<>b9
    proof
      assume
A35:  a9=b9;
A36:  now
        assume a9=c9;
        then b9=o by A2,A3,A5,A6,A27,A23,A28,A35,AFF_1:18;
        hence contradiction by A18,AFF_1:7;
      end;
      a,c // b,c or a9=c9 by A11,A12,A35,AFF_1:5;
      then c,a // c,b by A36,AFF_1:4;
      then LIN c,a,b by AFF_1:def 1;
      then LIN a,c,b by AFF_1:6;
      then a,c // a,b by AFF_1:def 1;
      then a,b // a,c by AFF_1:4;
      then a,c // K by A8,A13,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A6,AFF_1:23;
    end;
    LIN b9,x,a9 by A14,A16,A22,A25,AFF_1:21;
    then LIN b9,c9,a9 by A18,A24,A33,A26,AFF_1:8;
    then b9,c9 // b9,a9 by AFF_1:def 1;
    then b9,c9 // a9,b9 by AFF_1:4;
    then b,c // a9,b9 by A11,A29,AFF_1:5;
    then a,b // b,c by A10,A34,AFF_1:5;
    then b,c // K by A8,A13,AFF_1:32;
    then c,b // K by AFF_1:34;
    hence contradiction by A2,A4,A20,AFF_1:23;
end;
