intros.
red in |- *.
red in H0.
inversion_clear H0.
exists x.
inversion_clear H1.
exists x0.
inversion_clear H0.
exists (Map_to_equal_subsets H x1).
split.
inversion_clear H1.
repeat red in H0.
repeat red in |- *.
intros.
apply (H0 i j); auto with algebra.
simpl in |- *.
red in |- *.
simpl in H3.
red in H3.
apply Trans with (subtype_elt (x1 i)); auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H x1 i));
auto with algebra.
apply Trans with (subtype_elt (x1 j)); auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H x1 j));
auto with algebra.
split.
inversion_clear H1.
inversion_clear H2.
auto.
inversion_clear H1.
inversion_clear H2.
apply Trans with (sum (mult_by_scalars x0 (Map_embed x1))); auto with algebra.
Qed.

Lemma lin_indep_comp :
forall X Y : part_set V, X =' Y in _ -> lin_indep X -> lin_indep Y.