
theorem Th32:
  for C being Category, D being Categorial Category, I1 being
  Indexing of C for I2 being Indexing of D for T being TargetCat of I1 st D is
  TargetCat of I1 holds I2*I1 = I2*(I1-functor(C,T))
proof
  let C be Category, D be Categorial Category;
  let I1 be Indexing of C;
  let I2 be Indexing of D;
  let T be TargetCat of I1;
  assume D is TargetCat of I1;
  then reconsider D9 = D as TargetCat of I1;
A1: Image (I1-functor(C,rng I1)) = rng I1 by Def12;
  Image (I1-functor(C,D9)) = rng I1 & Image (I1-functor(C,T)) = rng I1 by Def12
;
  hence thesis by A1,Th11,Th22;
end;
