unfold lin_dep in |- *.
intros.
inversion_clear H0.
exists x.
inversion_clear H1.
exists x0.
inversion_clear H0.
exists (Map_include_map _ H x1).
split.
red in |- *.
inversion_clear H1.
inversion_clear H2.
red in H0.
unfold not in |- *.
intros.
unfold not in H0.
apply (H0 i j); auto with algebra.
split.
inversion_clear H1.
inversion_clear H2.
assumption.
inversion_clear H1.
inversion_clear H2.
apply Trans with (sum (mult_by_scalars x0 (Map_embed x1))); auto with algebra.
Qed.

(** - %\intoc{Corollary to theorem 1.6}% *)
Lemma lin_indep_include :
forall S1 S2 : part_set V, included S1 S2 -> lin_indep S2 -> lin_indep S1.