Section a_nice_fact_on_spans.
intros.
red in |- *.
simpl in |- *.
red in H.
intros.
generalize (span_ind_eqv_span S x).
unfold iff in |- *.
intros (LC1, LC2).
generalize (LC1 H0).
intro.
inversion H1.
generalize H2; clear H2.
cut (is_subspace W).
unfold is_subspace in |- *.
intros (W1, (W2, W3)).
2: apply is_subspace_OK; auto with algebra.
clear H0 H1 LC1 LC2.
generalize x.
clear x.
elim x0.
simpl in |- *.
intros.
apply in_part_comp_l with (zero V); auto with algebra.
simpl in |- *.
intros.
apply in_part_comp_l with (subtype_elt c); auto with algebra.
intros.
simpl in H2.
apply in_part_comp_l with (span_ind_injection s +' span_ind_injection s0);
auto with algebra.
intros.
simpl in H2.
apply in_part_comp_l with (c mX span_ind_injection s); auto with algebra.
Qed.

(* and of course the obvious *)
Lemma set_included_in_its_span :
forall (F : field) (V : vectorspace F) (S : part_set V), included S (span S).