
theorem Th26:
  for C,D,E being Category, F being Functor of C,D for G being
  Functor of D,E, I being Indexing of E holds (I*G)*F = I*(G*F)
proof
  let C,D,E be Category;
  let F be Functor of C,D;
  let G be Functor of D,E;
  let I be Indexing of E;
  set T = rng I;
  reconsider T9 = T as TargetCat of I*G by Th24;
  I*G = ((I-functor(E,T))*G)-indexing_of D by Th23;
  then (I*G)-functor(D,T9) = (I-functor(E,T))*G by Th18;
  hence (I*G)*F = ((I-functor(E,T))*G*F)-indexing_of C by Th23
    .= ((I-functor(E,T))*(G*F))-indexing_of C by RELAT_1:36
    .= I*(G*F) by Th23;
end;
