Section more.
Variable F : field.
Variable V : vectorspace F.
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.
intros S1 S2 H.
unfold lin_indep in |- *.
unfold not in |- *.
intros.
apply H0.
apply (lin_dep_include H).
assumption.
Qed.

(** For this we need $\neg\neg P\to P$: *)
Lemma lin_indep_prop :
forall (n : Nat) (a : seq n F) (v : seq n V),
distinct v ->
lin_indep (seq_set v) ->
sum (mult_by_scalars a v) =' (zero V) in _ -> a =' const_seq n (zero F) in _.