intros q1 qd1 qa1 d1 q2 qd2 qa2 d2 e1 e2 w H_aut1 H_aut2 H.
elim H.
intros e0 e3 H0 H1 H2 H3.
absurd (dans e3 (map injdroite q2)); auto.
apply absurd_injg_injd with q1.
rewrite <- H1; assumption.
intros e0 e e3 x w0 H0 H1 H2 H3 H4 H5 H6.
cut (dans e (map injgauche q1)).
2: apply transition_a_gauche with qa1 d1 q2 qd2 d2 e0 x; auto.
intro H7.
elim (H1 H7 H6); clear H1.
intros x1 Ht; elim Ht; clear Ht; intros x2 Ht; elim Ht; clear Ht;
intros w1' Ht; elim Ht; clear Ht; intros w2 Ht; elim Ht;
clear Ht; intros H8 Ht; elim Ht; clear Ht; intros H9 Ht;
elim Ht; clear Ht; intros H10 Ht; elim Ht; clear Ht;
intros H11 H12.
exists x1.
exists x2.
exists (cons x w1').
exists w2.
split; [ assumption | split; [ assumption | split ] ].
2: split.
2: assumption.
2: rewrite H12; auto.
apply chemin_cons with (first e); auto.
apply dans_map_injg; auto.
apply transition_a_gauche_2 with q1 qa1 q2 qd2 d2; auto.
intros e0 e e3 w0 H0 H1 H2 H3 H4 H5.
clear H1.
exists (first e0).
exists (first e).
exists nil.
exists w0.
cut (dans e0 (map injgauche qa1) /\ dans e (map injdroite qd2)).
2: apply transition_dans_pont with q1 d1 q2 d2; auto.
intro Ht; elim Ht; clear Ht; intros H6 H7.
split; [ apply dans_map_injg; assumption | split ].
apply dans_map_injd; assumption.
split.
apply chemin_nil; auto.
apply dans_trans with qa1.
apply dans_map_injg; auto.
apply automate_def3 with qd1 d1; auto.
split; auto.
apply chemin_A_chemin_2 with q1 qa1 d1 qd2; auto.
apply dans_map_trans with qd2; auto.
apply automate_def2 with qa2 d2; auto.
Qed.
(*  Si l1 reconnait w1 et l2 reconnait w2 alors l'automate construit 	*)
(*  ici reconnait (Append w1 w2).					*)

Lemma reconnait_Append :
forall (q1 qd1 qa1 d1 q2 qd2 qa2 d2 : Ensf) (w2 : Word)
(x1 x2 e2 : Elt) (w1 : Word) (e1 : Elt),
automate q1 qd1 qa1 d1 ->
automate q2 qd2 qa2 d2 ->
chemin e1 x1 q1 d1 w1 ->
dans x1 qa1 ->
chemin x2 e2 q2 d2 w2 ->
dans x2 qd2 ->
dans e2 qa2 ->
chemin_A (union_disj q1 q2)
(union (pont qa1 qd2) (union (injg_d1 q1 d1) (injd_d2 q2 d2)))
(couple e1 zero) (couple e2 un) (Append w1 w2).