intros.
red in H.
simpl in H.
red in H.
generalize (H x); intro.
inversion_clear H1.
simpl in H3.
auto.
Qed.

Lemma generates_then_is_lin_comb :
forall (F : field) (V : vectorspace F) (S W : part_set V),
generates S W -> forall x : V, in_part x W -> is_lin_comb x S.