intros.
inversion_clear H.
inversion_clear H0.
assert (has_n_elements m x).
apply all_finite_bases_equally_big with x0; auto with algebra.
generalize has_n_elements_inj; intro p.
apply (p _ _ _ H1 _ _ H0); auto with algebra.
Qed.

Lemma has_dim_easy :
forall (F : field) (V : vectorspace F) (b : part_set V),
is_basis b -> forall n : nat, has_n_elements n b -> has_dim n V.