
theorem Th34:
  for C,D being Category, F being Functor of C,D, I being Indexing
  of D for T being TargetCat of I, E being Categorial Category for G being
  Functor of T,E holds (G*I)*F = G*(I*F)
proof
  let C,D be Category, F be Functor of C,D, I be Indexing of D;
  let T be TargetCat of I;
  reconsider T9 = T as TargetCat of I*F by Th24;
  let E be Categorial Category, G be Functor of T,E;
  reconsider G9 = G as Functor of T9, E;
  reconsider GI = rng (G*I) as TargetCat of (G*(I-functor(D,T)))-indexing_of D
  by Def17;
A1: I*F = ((I-functor(D,T))*F)-indexing_of C by Th23;
A2: ((G*(I-functor(D,T)))-indexing_of D)-functor(D,GI) = G*(I-functor(D,T))
  by Th18;
  G*I = (G*(I-functor(D,T)))-indexing_of D & Image F is Subcategory of D
  by Def17;
  hence (G*I)*F = (((G*(I-functor(D,T)))-indexing_of D)-functor(D,GI)*F)
  -indexing_of C by Def16
    .= ((G*(I-functor(D,T))) * F) -indexing_of C by A2,Th2
    .= (G * ((I-functor(D,T))*F)) -indexing_of C by RELAT_1:36
    .= (G9*((I*F)-functor(C,T9)))-indexing_of C by A1,Th18
    .= G*(I*F) by Def17;
end;
