simpl in |- *.
red in |- *.
split; intro.
apply in_part_comp_r with (span_ind (span S):part_set V); auto with algebra.
change (Pred_fun (span S) x) in H.
simpl in |- *.
exists (Immediately (Build_subtype H)).
simpl in |- *.
apply Refl.
apply Trans with (span (span S):part_set V); auto with algebra.
assert (in_part x (span_ind (span S))).
apply in_part_comp_r with (span (span S):part_set V); auto with algebra.
elim H0.
intro.
generalize x.
elim x0.
simpl in |- *.
intros.
red in |- *.
exists 0.
exists (empty_seq F).
exists (empty_seq S).
simpl in |- *.
auto.
simpl in |- *.
intro.
elim c.
intros c0 cp.
elim cp.
intros x1 H1.
inversion H1.
inversion H2.
intros.
red in |- *.
exists x1.
exists x2.
exists x3.
apply Trans with c0; auto with algebra.
intros.
clear H0 H x x0.
simpl in H3.
simpl in |- *.
red in |- *.
generalize (H1 (span_ind_injection s) (Refl _)).
generalize (H2 (span_ind_injection s0) (Refl _)).
intros.
clear H1 H2.
simpl in H, H0.
red in H, H0.
inversion H.
inversion H0.
exists (x0 + x).
inversion H1.
inversion H2.
exists (x3 ++ x2).
inversion H4.
inversion H5.
exists (x5 ++ x4).
apply
Trans
with
(sum (mult_by_scalars x3 (Map_embed x5)) +'
sum (mult_by_scalars x2 (Map_embed x4))).
apply Trans with (span_ind_injection s +' span_ind_injection s0);
auto with algebra.
apply
Trans
with
(sum
(mult_by_scalars x3 (Map_embed x5) ++
mult_by_scalars x2 (Map_embed x4))).
apply Sym.
apply sum_concat; auto with algebra.
apply
Trans with (sum (mult_by_scalars (x3 ++ x2) (Map_embed x5 ++ Map_embed x4))).
unfold mult_by_scalars in |- *.
apply sum_comp; auto with algebra.
unfold mult_by_scalars in |- *.
apply sum_comp.
apply toMap.
apply pointwise_comp; auto with algebra.
intros.
clear x0 H0 H x.
simpl in H2.
generalize (H1 (span_ind_injection s) (Refl _)).
simpl in |- *.
unfold is_lin_comb in |- *.
intro.
inversion H.
inversion H0.
inversion H3.
exists x.
exists (pointwise (uncurry (RING_comp (R:=F))) (const_seq x c) x0).
exists x2.
apply Trans with (c mX span_ind_injection s); auto with algebra.
apply Trans with (c mX sum (mult_by_scalars x0 (Map_embed x2)));
auto with algebra.
apply
Trans
with
(sum
(mult_by_scalars (const_seq x c) (mult_by_scalars x0 (Map_embed x2))));
auto with algebra.
Qed.

Lemma span_preserves_inclusion :
forall S S' : part_set V, included S S' -> included (span S) (span S').