split.
2: apply max_lin_indep_subsets_of_generating_sets_are_bases;
auto with algebra.
2: red in |- *.
2: apply included_antisym; auto with algebra.
intro.
red in |- *.
split.
red in |- *.
simpl in |- *.
auto.
split.
red in H.
inversion_clear H.
auto.
red in H.
inversion_clear H.
intros.
elim (lin_dep_vs_span_lemma H1 H2).
intros.
apply H4; auto with algebra.
red in H0.
apply in_part_comp_r with (full V); auto with algebra.
Qed.

Lemma basis_is_max_lin_indep :
forall (F : field) (V : vectorspace F) (beta : basis V),
max_lin_indep beta (full V).