intros.
assert (in_part x (span W)).
simpl in |- *.
auto.
apply in_part_comp_r with (span W:part_set V).
auto.
elim (subspace_span_characterisation W).
intros.
apply H1.
apply is_subspace_OK.
Qed.

Lemma span_intersect :
forall S S' : part_set V,
included (span (inter S S')) (inter (span S) (span S')).