reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;

theorem Th3:
  for F1,F2 being Functor of A,B, G1,G2 being Functor of B,C st F1
  is_transformable_to F2 & G1 is_transformable_to G2 holds G1*F1
  is_transformable_to G2*F2
proof
  let F1,F2 be Functor of A,B, G1,G2 be Functor of B,C such that
A1: F1 is_transformable_to F2 and
A2: G1 is_transformable_to G2;
  let a be Object of A;
  Hom(F1.a,F2.a) <> {} by A1;
  then
A3: Hom(G1.(F1.a),G1.(F2.a)) <> {} by CAT_1:84;
A4: G1.(F1.a) = (G1*F1).a & G2.(F2.a) = (G2*F2).a by CAT_1:76;
  Hom(G1.(F2.a),G2.(F2.a)) <> {} by A2;
  hence thesis by A4,A3,CAT_1:24;
end;
