theorem Th35:
  F1, F2 are_naturally_equivalent implies G1*F1, G1*F2
  are_naturally_equivalent & G1*e is natural_equivalence of G1*F1, G1*F2
proof
  assume
A1: F1, F2 are_naturally_equivalent;
  then
A2: F2 is_transformable_to F1;
A3: F1 is_naturally_transformable_to F2 by A1;
  then reconsider k = G1*e as natural_transformation of G1*F1, G1*F2 by Th28;
A4: F1 is_transformable_to F2 by A1,Def4;
A5: now
    let a be Object of A;
A6: (G1*F1).a = G1.(F1.a) & (G1*F2).a = G1.(F2.a) by FUNCTOR0:33;
A7: <^F2.a,F1.a^> <> {} by A2;
    k!a = G1.(e!a) & <^F1.a,F2.a^> <> {} by A4,Th11;
    hence k!a is iso by A1,A6,A7,Def5,ALTCAT_4:20;
  end;
  G1*F1, G1*F2 are_naturally_equivalent
  by A3,Lm2,A2,Th10,A5;
  hence thesis by A5,Def5;
end;
