
theorem Th6:
  for A,B being category, F,F1,F2,F3 being Functor of A,B
  holds F is_transformable_to F1 & F1 is_transformable_to F2 & F2
  is_transformable_to F3 implies for t1 being transformation of F,F1, t2 being
transformation of F1,F2, t3 being transformation of F2,F3 holds t3`*`t2`*`t1 =
  t3`*`(t2`*`t1)
proof
  let A,B be category, F,F1,F2,F3 be Functor of A,B;
  assume that
A1: F is_transformable_to F1 and
A2: F1 is_transformable_to F2 and
A3: F2 is_transformable_to F3;
  let t1 be transformation of F,F1, t2 be transformation of F1,F2, t3 be
  transformation of F2,F3;
A4: F1 is_transformable_to F3 by A2,A3,Th2;
A5: F is_transformable_to F2 by A1,A2,Th2;
  now
    let a be Object of A;
A6: <^F2.a,F3.a^> <> {} by A3;
A7: <^F.a,F1.a^> <> {} & <^F1.a,F2.a^> <> {} by A1,A2;
    thus (t3`*`t2`*`t1)!a = ((t3`*`t2)!a)*(t1!a) by A1,A4,Def5
      .= (t3!a)*(t2!a)*(t1!a) by A2,A3,Def5
      .= (t3!a)*((t2!a)*(t1!a)) by A7,A6,ALTCAT_1:21
      .= (t3!a)*((t2`*`t1)!a) by A1,A2,Def5
      .= (t3`*`(t2`*`t1))!a by A3,A5,Def5;
  end;
  hence thesis by A1,A4,Th2,Th3;
end;
