intros.
red in H0.
red in |- *.
apply Trans with (span S:part_set _); auto with algebra.
apply included_antisym; auto with algebra.
apply included_trans with (full V); auto with algebra.
Qed.

Lemma subspace_span_characterisation :
forall W : part_set V, is_subspace W <-> span W =' W in part_set V.