simple induction w.
intro.
apply lwordnil_is_regS.
intros a w0 H H4.
cut (inmonoid alph w0).
2: apply (inmonoid_cons_inv alph w0 a); auto.
intro H5.
cut
(exists q : Ensf,
(exists e : Elt,
(exists qa : Ensf,
(exists d : Ensf,
automate q (singleton e) qa d /\
eqwordset (reconnait q (singleton e) qa d) (lword w0)))));
auto.
clear H; intro H; elim H; clear H.
intros q H0; elim H0; clear H0.
intros e H0; elim H0; clear H0.
intros qa H0; elim H0; clear H0.
intros d H0; elim H0; clear H0.
intros.
cut (exists e0 : Elt, ~ dans e0 q).
2: apply exist_other; auto.
intro H2; elim H2; clear H2.
intros e0 H2.
exists (add e0 q).
exists e0.
exists qa.
exists (add (couple e0 (couple a e)) d).
elim H; clear H.
intros H H1.
elim H1; clear H1.
intros.
split.
red in |- *.
split; auto.
split; auto.
apply add_inclus.
apply coupl2_inv; auto.
apply coupl2_inv.
apply (inmonoid_cons_inv2 alph a w0); auto.
cut (dans e q); auto.
apply (inclus_trans d (prodcart q (prodcart alph q))); auto.
apply auto_cons; auto.
apply (inmonoid_cons_inv2 alph a w0); auto.
unfold automate in |- *; auto.
Qed.
(*									*)
(*  Finalement, on montre qu'un langage reduit a un mot est regulier : 	*)
(*									*)

Lemma lword_is_reg : forall w : Word, inmonoid alph w -> isregular (lword w).
unfold isregular in |- *.