Fixpoint v2k (n : nat) : vect n -> kn n :=
match n return vect n -> kn n  with
| O => fun v : vect 0 => tt
| S n1  => fun v => let (v1,v2) := v in
('C[v1], v2k n1 v2)
end.

(* Anti-commutativity for homegeonous vectors, generalization of  cbl_base_prod_com  *)
Lemma join_hom_com n k1 k2 x y :
hom n k1 x -> hom n k2 y ->  y ∨ x = ((- (1)) ^(k1 * k2)).* (x ∨ y).