intros.
red in |- *.
red in |- *.
red in H0.
red in H0.
intro.
apply H0.
apply lin_dep_comp with Y; auto with algebra.
Qed.

Lemma lin_dep_defs_eqv : forall X : part_set V, lin_indep X <-> lin_indep' X.
unfold lin_indep in |- *.