Hint Resolve join_hom.

(* Specifications: *)
Lemma join_big n k1 k2 (x y : vect n) :
hom n k1 x -> hom n k2 y -> n < k1 + k2 -> x ∨ y = 0.
Lemma lift_decomp n (x y: vect n) : (x, y) = 'e_0 ∨ x^'l + y^'l.
(* Base as product *)
Lemma base_in n k x: In x (base n k.+1) ->
exists i, exists y, x = 'e_i ∨ y /\ In y (base n k).