intros; red in |- *; red in H; intro.
red in H; (apply H; auto with algebra).
apply zero_imp_lin_dep; auto with algebra.
Qed.

(* Specifications: *)
Lemma inject_subsets_lin_dep :
forall (W : subspace V) (X : part_set W),
lin_dep X <-> lin_dep (inject_subsets X).