theorem Th36:
  G1, G2 are_naturally_equivalent implies G1*F1, G2*F1
  are_naturally_equivalent & f*F1 is natural_equivalence of G1*F1, G2*F1
proof
  assume
A1: G1, G2 are_naturally_equivalent;
  then
 G1 is_naturally_transformable_to G2;
  then reconsider k = f*F1 as natural_transformation of G1*F1, G2*F1 by Th29;
A2: now
    let a be Object of A;
    G1 is_transformable_to G2 by A1,Def4;
    then
A3: k!a = f!(F1.a) by Th12;
    (G1*F1).a = G1.(F1.a) & (G2*F1).a = G2.(F1.a) by FUNCTOR0:33;
    hence k!a is iso by A1,A3,Def5;
  end;
  G1*F1, G2*F1 are_naturally_equivalent
  by Lm2,A1,Th10,A2;
  hence thesis by A2,Def5;
end;
