intros.
red in |- *; red in H1.
apply Trans with (span S:part_set V); auto with algebra.
apply Trans with (W:part_set V); auto with algebra.
Qed.

Lemma generated_subsets_are_subspaces :
forall (F : field) (V : vectorspace F) (S W : part_set V),
generates S W -> is_subspace W.