intros.
apply Trans with (span S:part_set V); auto with algebra.
apply Sym; (apply span_is_span_ind; auto with algebra).
apply Trans with (span S':part_set V); auto with algebra.
apply span_is_span_ind; auto with algebra.
Qed.
End spans_eqv.
Hint Resolve span_is_span_ind span_ind_comp: algebra.
Section a_nice_fact_on_spans.

(** - %\intoc{Theorem 1.5 (cont'd)}% *)
Lemma span_smallest_subspace_containing_subset :
forall (F : field) (V : vectorspace F) (W : subspace V) (S : part_set V),
included S W -> included (span S) W.