intros.
red in H.
elim H; clear H; intros.
elim H0; clear H0; intros.
case (classic (in_part (subtype_elt w) W')).
intro; red in |- *.
exists 1.
exists (const_seq (A:=F) 1 one).
exists (const_seq (A:=W') 1 (Build_subtype H2)).
simpl in |- *.
apply Trans with (one mX subtype_elt w); auto with algebra.
intro.
generalize (lin_dep_vs_span_lemma (F:=F) (V:=V) (s:=W')).
intros.
generalize (H3 H0 (subtype_elt w) H2).
intro.
clear H3; inversion_clear H4.
apply H3; auto with algebra.
Qed.

(* Specifications: *)
(** - But of course we do have span(W')=span(W): *)
Lemma max_lin_indep_subset_has_same_span :
forall (F : field) (V : vectorspace F) (W W' : part_set V),
max_lin_indep W' W -> span W =' span W' in part_set V.