theorem Th28:
  F1 is_naturally_transformable_to F2 implies G1*t is
  natural_transformation of G1*F1, G1*F2
proof
  assume
A1: F1 is_naturally_transformable_to F2;
  then
A2: F1 is_transformable_to F2;
  thus G1*F1 is_naturally_transformable_to G1*F2 by A1,Lm2;
  let a, b be Object of A such that
A3: <^a,b^> <> {};
A4: (G1*F1).b = G1.(F1.b) & <^F1.a,F1.b^> <> {} by A3,FUNCTOR0:33,def 18;
A5: <^F1.b,F2.b^> <> {} by A2;
A6: <^F1.a,F2.a^> <> {} by A2;
  reconsider G1ta = G1.(t!a) as Morphism of G1.(F1.a), (G1*F2).a by FUNCTOR0:33
;
  reconsider G1tb = G1.(t!b) as Morphism of (G1*F1).b, G1.(F2.b) by FUNCTOR0:33
;
  let f be Morphism of a, b;
A7: (G1*F2).a = G1.(F2.a) by FUNCTOR0:33;
A8: <^F2.a,F2.b^> <> {} by A3,FUNCTOR0:def 18;
A9: (G1*F1).a = G1.(F1.a) by FUNCTOR0:33;
  then reconsider G1F1f = G1.(F1.f) as Morphism of (G1*F1).a, (G1*F1).b by
FUNCTOR0:33;
A10: (G1*F2).b = G1.(F2.b) by FUNCTOR0:33;
  hence (G1*t)!b*(G1*F1).f = G1tb*((G1*F1).f) by A2,Th11
    .= G1tb*G1F1f by A3,Th6
    .= G1.(t!b*F1.f) by A9,A4,A5,FUNCTOR0:def 23
    .= G1.(F2.f*(t!a)) by A1,A3,FUNCTOR2:def 7
    .= G1.(F2.f)*G1.(t!a) by A6,A8,FUNCTOR0:def 23
    .= (G1*F2).f*G1ta by A3,A7,A10,Th6
    .= (G1*F2).f*((G1*t)!a) by A2,A9,Th11;
end;
