intros.
red in |- *.
split.
unfold union_disj in |- *.
apply inclus_d2.
apply map_inclus.
apply automate_def3 with qd2 d2; auto.
split.
unfold union_disj in |- *.
apply inclus_g2.
apply map_inclus.
apply automate_def2 with qa1 d1; auto.
apply union_inclus.
unfold pont in |- *.
unfold inclus in |- *.
intros.
cut
(exists y : Elt, dans y (prodcart qa1 qd2) /\ x = transition_pont y :>Elt).
2: apply dans_map; auto.
intro Ht; elim Ht; clear Ht.
intros y Ht; elim Ht; clear Ht; intros H2 H3.
cut
(exists y1 : Elt,
(exists y2 : Elt, dans y1 qa1 /\ dans y2 qd2 /\ y = couple y1 y2 :>Elt)).
2: apply coupl3; auto.
intro Ht; elim Ht; clear Ht.
intros y1 Ht; elim Ht; clear Ht.
intros y2 Ht; elim Ht; clear Ht.
intros H4 Ht; elim Ht; clear Ht; intros H5 H6.
cut (x = couple (couple y1 zero) (couple epsilon (couple y2 un)) :>Elt).
2: rewrite H3.
2: rewrite H6; auto.
intro H7.
rewrite H7.
apply coupl2_inv.
unfold union_disj in |- *.
apply union_g.
replace (couple y1 zero) with (injgauche y1); auto.
apply dans_map_inv.
apply dans_trans with qa1; auto.
apply automate_def3 with qd1 d1; auto.
apply coupl2_inv; auto.
unfold union_disj in |- *.
apply union_d.
replace (couple y2 un) with (injdroite y2); auto.
apply dans_map_inv.
apply dans_trans with qd2; auto.
apply automate_def2 with qa2 d2; auto.
apply union_inclus.
apply
inclus_trans
with (prodcart (map injgauche q1) (prodcart alph (map injgauche q1))).
unfold injg_d1 in |- *.
apply inclus_tq.
unfold union_disj in |- *.
apply cart_inclus.
apply inclus_g.
apply cart_inclus; auto.
apply
inclus_trans
with (prodcart (map injdroite q2) (prodcart alph (map injdroite q2))).
unfold injd_d2 in |- *.
apply inclus_tq.
unfold union_disj in |- *.
apply cart_inclus; auto.
Qed.
(*  Si on a une transition (e0,x,e) avec e0 dans (map injgauche q1)	*)
(*  et x dans alph, alors e est dans (map injgauche q1).		*)

Lemma transition_a_gauche :
forall (q1 qa1 d1 q2 qd2 d2 : Ensf) (e0 x e : Elt),
dans e0 (map injgauche q1) ->
dans x alph ->
dans (couple e0 (couple x e))
(union (pont qa1 qd2) (union (injg_d1 q1 d1) (injd_d2 q2 d2))) ->
dans e (map injgauche q1).