
theorem
  for C being Category, D being Categorial Category, I1 being Indexing
  of C for I2 being Indexing of D for T being TargetCat of I2 st D is TargetCat
  of I1 holds I2*I1 = (I2-functor(D,T))*I1
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 I2;
  assume D is TargetCat of I1;
  then reconsider D9 = D as TargetCat of I1;
  reconsider I29 = I2 as Indexing of D9;
  reconsider T9 = T as TargetCat of I29;
  Image (I1-functor(C,D9)) = rng I1 & Image (I1-functor(C,rng I1)) = rng
  I1 by Def12;
  hence I2*I1 = I2*(I1-functor(C,D9)) by Th11,Th22
    .= (I29-functor(D9,T9)*(I1-functor(C,D9)))-indexing_of C by Th23
    .= (I2-functor(D,T))*I1 by Def17;
end;
