elim every_finite_generating_set_has_a_subset_that_is_a_basis;
auto with algebra.
intros.
exists (Build_basis H1).
simpl in |- *.
red in |- *; simpl in |- *.
intros.
inversion_clear H2.
apply in_part_comp_l with (subtype_elt (subtype_elt x1)); auto with algebra.
Qed.

Lemma every_finitely_generated_vectorspace_has_a_finite_basis :
exists W : part_set V, is_finite_subset W /\ is_basis W.