theorem
  F1, F2 are_naturally_equivalent & G1, G2 are_naturally_equivalent
implies G1*F1, G2*F2 are_naturally_equivalent & f (#) e is natural_equivalence
  of G1*F1, G2*F2
proof
  assume that
A1: F1, F2 are_naturally_equivalent and
A2: G1, G2 are_naturally_equivalent;
A3: G1*F1, G1*F2 are_naturally_equivalent by A1,Th35;
  G1 is_naturally_transformable_to G2 by A2;
  then reconsider sF2 = f*F2 as natural_transformation of G1*F2, G2*F2 by Th29;
  F1 is_naturally_transformable_to F2 by A1;
  then reconsider G1t = G1*e as natural_transformation of G1*F1, G1*F2 by Th28;
A4: G1*F2, G2*F2 are_naturally_equivalent by A2,Th36;
  f*F2 is natural_equivalence of G1*F2, G2*F2 & G1*e is
  natural_equivalence of G1*F1, G1*F2 by A1,A2,Th35,Th36;
  then sF2`*`G1t is natural_equivalence of G1*F1, G2*F2 by A4,A3,Th34;
  hence thesis by A4,A3,Th33,FUNCTOR2:def 8;
end;
