reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;
reserve AFP for AffinPlane,
  a,a9,b,b9,c,c9,d,p,p9,q,q9,r,x,x9,y,y9,z for Element of AFP,
  A,C,P for Subset of AFP,
  f,g,h,f1,f2 for Permutation of the carrier of AFP;

theorem Th4:
  AFP is translational iff for a,a9,b,c,b9,c9 st not LIN a,a9,b &
  not LIN a,a9,c & a,a9 // b,b9 & a,a9 // c,c9 & a,b // a9,b9 & a,c // a9,c9
  holds b,c // b9,c9
proof
  thus AFP is translational implies for a,a9,b,c,b9,c9 st not LIN a,a9,b & not
LIN a,a9,c & a,a9 // b,b9 & a,a9 // c,c9 & a,b // a9,b9 & a,c // a9,c9 holds b,
  c // b9,c9
  proof
    assume
A1: AFP is translational;
    let a,a9,b,c,b9,c9;
    assume that
A2: not LIN a,a9,b and
A3: not LIN a,a9,c and
A4: a,a9 // b,b9 and
A5: a,a9 // c,c9 and
A6: a,b // a9,b9 and
A7: a,c // a9,c9;
    set A=Line(a,a9), P=Line(b,b9), C=Line(c,c9);
A8: a in A & a9 in A by AFF_1:15;
A9: c <>c9
    proof
      assume c = c9;
      then c,a // c,a9 by A7,AFF_1:4;
      then LIN c,a,a9 by AFF_1:def 1;
      hence contradiction by A3,AFF_1:6;
    end;
    then
A10: C is being_line by AFF_1:def 3;
A11: b<>b9
    proof
      assume b=b9;
      then b,a // b,a9 by A6,AFF_1:4;
      then LIN b,a,a9 by AFF_1:def 1;
      hence contradiction by A2,AFF_1:6;
    end;
    then
A12: P is being_line by AFF_1:def 3;
A13: a<>a9 by A2,AFF_1:7;
    then
A14: A is being_line by AFF_1:def 3;
A15: c in C by AFF_1:15;
    then
A16: A<>C by A3,A8,A14,AFF_1:21;
A17: A // P by A4,A13,A11,AFF_1:37;
A18: b9 in P & c9 in C by AFF_1:15;
A19: A // C by A5,A13,A9,AFF_1:37;
A20: b in P by AFF_1:15;
    then A<>P by A2,A8,A14,AFF_1:21;
    hence thesis by A1,A6,A7,A8,A20,A15,A18,A14,A12,A10,A17,A19,A16,
AFF_2:def 11;
  end;
  assume
A21: for a,a9,b,c,b9,c9 st not LIN a,a9,b & not LIN a,a9,c & a,a9 // b,
  b9 & a,a9 // c,c9 & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9;
  now
    let A,P,C,a,b,c,a9,b9,c9;
    assume that
A22: A // P and
A23: A // C and
A24: a in A and
A25: a9 in A and
A26: b in P and
A27: b9 in P and
A28: c in C and
A29: c9 in C and
A30: A is being_line and
A31: P is being_line and
A32: C is being_line and
A33: A<>P and
A34: A<>C and
A35: a,b // a9,b9 and
A36: a,c // a9,c9;
A37: a,a9 // b,b9 & a,a9 // c,c9 by A22,A23,A24,A25,A26,A27,A28,A29,AFF_1:39;
A38: now
      assume
A39:  a<>a9;
A40:  not LIN a,a9,c
      proof
        assume LIN a,a9,c;
        then c in A by A24,A25,A30,A39,AFF_1:25;
        hence contradiction by A23,A28,A34,AFF_1:45;
      end;
      not LIN a,a9,b
      proof
        assume LIN a,a9,b;
        then b in A by A24,A25,A30,A39,AFF_1:25;
        hence contradiction by A22,A26,A33,AFF_1:45;
      end;
      hence b,c // b9,c9 by A21,A35,A36,A37,A40;
    end;
    now
      assume
A41:  a=a9;
      then LIN a,c,c9 by A36,AFF_1:def 1;
      then LIN c,c9,a by AFF_1:6;
      then
A42:  c = c9 or a in C by A28,A29,A32,AFF_1:25;
      LIN a,b,b9 by A35,A41,AFF_1:def 1;
      then LIN b,b9,a by AFF_1:6;
      then b=b9 or a in P by A26,A27,A31,AFF_1:25;
      hence b,c // b9,c9 by A22,A23,A24,A33,A34,A42,AFF_1:2,45;
    end;
    hence b,c // b9,c9 by A38;
  end;
  hence thesis by AFF_2:def 11;
end;
