intros.
apply included_antisym.
apply span_smallest_subspace_containing_subset; auto with algebra.
apply included2_union; auto with algebra.
apply included_trans with (union (span S) (span S')); auto with algebra.
apply included_trans with (span S:part_set V); auto with algebra.
apply included_trans with (union (span S) (span S')); auto with algebra.
apply included_trans with (span S':part_set V); auto with algebra.
apply span_smallest_subspace_containing_subset; auto with algebra.
Qed.

Lemma inclusion_generates :
forall S S' : part_set V,
included S S' -> generates S (full V) -> generates S' (full V).