reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th18:
  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
  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,Th14;
A5: F is_transformable_to F2 by A1,A2,Th14;
  now
    let a be Object of A;
A6: Hom(F.a,F1.a) <> {} by A1;
A7: Hom(F1.a,F2.a) <> {} by A2;
A8: Hom(F2.a,F3.a) <> {} by A3;
    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 A6,A7,A8,CAT_1:25
      .= (t3.a)*((t2`*`t1).a) by A1,A2,Def5
      .= (t3`*`(t2`*`t1)).a by A3,A5,Def5;
  end;
  hence thesis by A1,A4,Th14,Th15;
end;
