theorem Th41:
  for F being Functor of A,B, G being Functor of C,D for I,J being
  Functor of B,C st I ~= J holds G*I ~= G*J & I*F ~= J*F
proof
  let F be Functor of A,B, G be Functor of C,D;
  let I,J be Functor of B,C;
  assume
A1: I is_naturally_transformable_to J;
  given t being natural_transformation of I,J such that
A2: t is invertible;
  thus G*I ~= G*J
  proof
    thus G*I is_naturally_transformable_to G*J by A1,Th20;
    take G*t;
    let b be Object of B;
A3: t.b is invertible by A2;
A4:  G.(I.b) = (G*I).b & G.(J.b) = (G*J).b by CAT_1:76;
  (G*t).b = G/.(t.b) by A1,Th21;
    hence (G*t).b is invertible by A3,Th1,A4;
  end;
  thus I*F is_naturally_transformable_to J*F by A1,Th20;
  take t*F;
  let a be Object of A;
A5:  I.(F.a) = (I*F).a & J.(F.a) = (J*F).a by CAT_1:76;
 (t*F).a = t.(F.a) by A1,Th22;
  hence (t*F).a is invertible by A2,A5;
end;
