intros q1 qd1 qa1 d1 q2 qd2 qa2 d2 w2 x1 x2 e2.
simple induction w1.
intros e1 H_aut1 H_aut2 H H0 H1 H2 H3.
cut (Chemin e1 x1 q1 d1 nil); auto.
intro H4.
cut (dans e1 q1 /\ e1 = x1 :>Elt); auto.
intro Ht; elim Ht; clear Ht; intros H5 H6.
replace (Append nil w2) with w2; auto.
apply chemin_A_epsilon with (couple x2 un).
apply chemin_A_d1_d2 with (union (injg_d1 q1 d1) (injd_d2 q2 d2));
auto.
apply chemin_chemin_A.
apply chemin_extension_2 with qd2 qa2; auto.
apply dans_trans with qd2; auto.
apply automate_def2 with qa2 d2; auto.
replace (couple e1 zero) with (injgauche e1); auto.
unfold union_disj in |- *.
apply union_g.
apply dans_map_inv; auto.
apply union_g.
unfold pont in |- *.
rewrite H6.
replace (couple (couple x1 zero) (couple epsilon (couple x2 un))) with
(transition_pont (couple x1 x2)); auto.
intros x w H e1 H0 H1 H2 H3 H4 H5 H6.
replace (Append (cons x w) w2) with (cons x (Append w w2)); auto.
cut (Chemin e1 x1 q1 d1 (cons x w)); auto.
intro H7.
cut
(exists e : Elt,
chemin e x1 q1 d1 w /\
dans e1 q1 /\ dans x alph /\ dans (couple e1 (couple x e)) d1);
auto.
intro Ht; elim Ht; clear Ht; intros e Ht; elim Ht; clear Ht; intros H8 Ht;
elim Ht; clear Ht; intros H9 Ht; elim Ht; clear Ht;
intros H10 H11.
apply chemin_A_cons with (couple e zero); auto.
unfold union_disj in |- *.
apply union_g.
replace (couple e1 zero) with (injgauche e1); auto.
apply union_d.
apply union_g.
unfold injg_d1 in |- *.
apply imp_dans_tq; auto.
apply coupl2_inv.
replace (couple e1 zero) with (injgauche e1); auto.
apply coupl2_inv; auto.
replace (couple e zero) with (injgauche e); auto.
apply dans_map_inv; auto.
apply dans_e1_q with d1 w x1; auto.
Qed.
(*  Si l'automate 1 reconnait l1 et si l'automate 2 reconnait l2	*)
(*  alors l'automate ci-dessous reconnait l1.l2 .			*)

Lemma lconc_is_reg1 :
forall (q1 qd1 qa1 d1 q2 qd2 qa2 d2 : Ensf) (l1 l2 : wordset),
automate q1 qd1 qa1 d1 ->
eqwordset (reconnait q1 qd1 qa1 d1) l1 ->
automate q2 qd2 qa2 d2 ->
eqwordset (reconnait q2 qd2 qa2 d2) l2 ->
eqwordset
(reconnait_A (union_disj q1 q2) (map injgauche qd1)
(map injdroite qa2)
(union (pont qa1 qd2) (union (injg_d1 q1 d1) (injd_d2 q2 d2))))
(lconc l1 l2).