Hint Resolve meet_hom.

(* One theorem of white  WHITE 1997 p882 and Barnabei Brini Rota p135 *)
Lemma join2_meetE n k1 k2 (x y : vect n) :
hom n k1 x -> hom n k2 y -> n <= k1 + k2 ->  x ∨ y = (x ∧ y) ∨ E.