(* Base of k vector *)
Fixpoint base (n: nat) k:list (vect n) :=
  match n return list (vect n) with 
  | O =>  match k with | O => 1%f::nil | _ => nil end
  | S n1 => 
          match k with | O => 
                     map (lift n1) (base n1 k) | S k1 => 
           (map (dlift n1) (base n1 k1) ++ 
            map (lift n1) (base n1 k))%list
          end
  end.

Lemma proj_homk n (x: vect n) k : hom n k x -> proj n k x *X* base n k = x.