intro.
split.
intros.
apply Trans with (span_ind W:part_set V); auto with algebra.
simpl in |- *.
red in |- *.
split.
intro.
red in H.
inversion_clear H.
inversion_clear H2.
simpl in H0.
inversion_clear H0.
generalize H2.
generalize x.
clear H2.
induction x0 as [| c| x1 IHx1 x2 IHx2| c x0 IHx0].
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.
simpl in |- *.
intros.
apply in_part_comp_l with (span_ind_injection x1 +' span_ind_injection x2);
auto with algebra.
intros.
simpl in H2.
apply in_part_comp_l with (c mX span_ind_injection x0); auto with algebra.
intro.
simpl in |- *.
red in H0.
exists (Immediately (Build_subtype H0)).
simpl in |- *.
apply Refl.
intro.
apply is_subspace_comp with (span W:part_set V); auto with algebra.
apply span_is_subspace; auto with algebra.
Qed.

Lemma subspace_contains_all_lin_combs :
forall (W : subspace V) (x : V), is_lin_comb x W -> in_part x W.