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
  AP is satisfying_TDES_2 implies AP is satisfying_TDES_3
proof
  assume
A1: AP is satisfying_TDES_2;
    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: not a in K and
A6: o<>c and
A7: a<>b and
A8: LIN o,a,a9 and
A9: LIN o,b,b9 and
A10: a,b // a9,b9 and
A11: a,c // a9,c9 and
A12: b,c // b9,c9 and
A13: a,b // K;
    set A=Line(o,a), P=Line(o,b), N=Line(b,c);
A14: o in A by A3,A5,AFF_1:24;
A15: not LIN a,b,c
    proof
      assume LIN a,b,c;
      then a,b // a,c by AFF_1:def 1;
      then a,c // K by A7,A13,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A5,AFF_1:23;
    end;
A16: o<>b by A3,A5,A13,AFF_1:35;
    then
A17: b in P by AFF_1:24;
A18: a9,b9 // b,a by A10,AFF_1:4;
A19: b<>c by A4,A5,A13,AFF_1:35;
    then
A20: b in N & c in N by AFF_1:24;
A21: a in A by A3,A5,AFF_1:24;
A22: A is being_line by A3,A5,AFF_1:24;
A23: A<>P
    proof
      assume A=P;
      then a,b // A by A22,A21,A17,AFF_1:40,41;
      hence contradiction by A3,A5,A7,A13,A14,A21,AFF_1:45,53;
    end;
    assume
A24: not c9 in K;
A25: P is being_line by A16,AFF_1:24;
A26: o in P by A16,AFF_1:24;
    then
A27: b9 in P by A9,A16,A25,A17,AFF_1:25;
A28: a9 in A by A3,A5,A8,A22,A14,A21,AFF_1:25;
A29: a9<>b9
    proof
      assume
A30:  a9=b9;
      then a,c // b,c or a9=c9 by A11,A12,AFF_1:5;
      then c,a // c,b or a9=c9 by AFF_1:4;
      then LIN c,a,b or a9=c9 by AFF_1:def 1;
      hence contradiction by A3,A24,A15,A22,A25,A14,A26,A28,A27,A23,A30,AFF_1:6
,18;
    end;
A31: a9<>c9
    proof
      assume a9=c9;
      then b,c // a9,b9 by A12,AFF_1:4;
      then a,b // b,c by A10,A29,AFF_1:5;
      then b,a // b,c by AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      hence contradiction by A15,AFF_1:6;
    end;
    not a9,c9 // K
    proof
      assume
A32:  a9,c9 // K;
      a9,c9 // a,c by A11,AFF_1:4;
      then a,c // K by A31,A32,AFF_1:32;
      then c,a // K by AFF_1:34;
      hence contradiction by A2,A4,A5,AFF_1:23;
    end;
    then consider x such that
A33: x in K and
A34: LIN a9,c9,x by A2,AFF_1:59;
    a9,c9 // a9,x by A34,AFF_1:def 1;
    then
A35: a,c // a9,x by A11,A31,AFF_1:5;
    N is being_line by A19,AFF_1:24;
    then consider T such that
A36: x in T and
A37: N // T by AFF_1:49;
A38: not b in K by A5,A13,AFF_1:35;
A39: not T // P
    proof
      assume T // P;
      then N // P by A37,AFF_1:44;
      then c in P by A17,A20,AFF_1:45;
      hence contradiction by A2,A3,A4,A6,A38,A25,A26,A17,AFF_1:18;
    end;
    T is being_line by A37,AFF_1:36;
    then consider y such that
A40: y in T and
A41: y in P by A25,A39,AFF_1:58;
A42: b,c // y,x by A20,A36,A37,A40,AFF_1:39;
A43: now
      assume y=b9;
      then b9,c9 // b9,x by A12,A19,A42,AFF_1:5;
      then LIN b9,c9,x by AFF_1:def 1;
      then
A44:  LIN c9,x,b9 by AFF_1:6;
      LIN c9,x,a9 & LIN c9,x,c9 by A34,AFF_1:6,7;
      then LIN a9,b9,c9 by A24,A33,A44,AFF_1:8;
      then a9,b9 // a9,c9 by AFF_1:def 1;
      then a9,b9 // a,c by A11,A31,AFF_1:5;
      then a,b // a,c by A10,A29,AFF_1:5;
      hence contradiction by A15,AFF_1:def 1;
    end;
    LIN o,b,y by A25,A26,A17,A41,AFF_1:21;
    then a,b // a9,y by A1,A2,A3,A4,A5,A6,A7,A8,A13,A33,A42,A35;
    then a9,b9 // a9,y by A7,A10,AFF_1:5;
    then LIN a9,b9,y by AFF_1:def 1;
    then LIN b9,y,a9 by AFF_1:6;
    then a9 in P by A25,A27,A41,A43,AFF_1:25;
    then a in P by A25,A17,A27,A29,A18,AFF_1:48;
    hence contradiction by A3,A5,A25,A26,A23,AFF_1:24;
end;
