red in |- *.
elim X.
clear X.
intros X H.
red in H.
inversion_clear H.
red in H0.
unfold span in H0.
unfold is_lin_comb in H0.
simpl in H0.
red in H0.
simpl in H0.
generalize (H0 x).
clear H0.
unfold is_lin_comb in |- *.
intros (H', H0).
apply H0; auto with algebra.
Qed.

Lemma basis_prop_strong :
forall (n : Nat) (v : seq n V),
is_basis (seq_set v) ->
forall x : V, exists a : seq n F, sum (mult_by_scalars a v) =' x in _.