
theorem Th28:
  for C being Category, I being Indexing of C for T being
  TargetCat of I, D being Categorial Category for F being Functor of T,D holds
  Image F is TargetCat of F*I
proof
  let C be Category;
  let I be Indexing of C;
  let T be TargetCat of I, D be Categorial Category;
  let F be Functor of T,D;
  reconsider F9 = F as Functor of T, Image F by CAT_5:8;
  set T9 = the TargetCat of F*I;
A1: rng (F*I) = Image ((F*I)-functor(C,T9)) by Def12;
  F*I = F9*I by Th27
    .= (F9*(I-functor(C,T)))-indexing_of C by Def17;
  then rng (F*I) = Image (F9*(I-functor(C,T))) by A1,Th18,CAT_5:22;
  hence thesis by Th14;
end;
