
theorem Th22:
  for C,D1,D2,E being Category, I being Indexing of E for F being
  Functor of C,D1 for G being Functor of C,D2 st Image F is Subcategory of E &
  Image G is Subcategory of E & F = G holds I*F = I*G
proof
  let C,D1,D2,E be Category, I be Indexing of E;
  let F be Functor of C,D1, G be Functor of C,D2;
  assume that
A1: Image F is Subcategory of E and
A2: Image G is Subcategory of E and
A3: F = G;
  reconsider F9 = F as Functor of C, Image F by CAT_5:8;
  reconsider F9 as Functor of C,E by A1,Lm6;
  I*F = (I-functor(E,rng I)*F9)-indexing_of C by A1,Def16;
  hence thesis by A2,A3,Def16;
end;
