intros q qa d g0 e0 x e H H0.
unfold fun_d_dstar in |- *.
intro.
cut
(dans (couple e0 (couple x e)) d \/
dans (couple e0 (couple x e)) (delta g0 qa)); auto.
intro Ht; elim Ht; clear Ht.
intro; assumption.
unfold delta in |- *.
intro.
cut
(exists y : Elt,
dans y qa /\ couple e0 (couple x e) = transition_back g0 y :>Elt).
2: apply dans_map; auto.
intro Ht; elim Ht; clear Ht; intros y Ht; elim Ht; clear Ht; intros.
unfold transition_back in H4.
cut (e0 = y :>Elt /\ couple x e = couple epsilon g0 :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
cut (x = epsilon :>Elt /\ e = g0 :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
absurd (dans x alph); auto.
rewrite H7.
red in |- *; intro; apply not_dans_epsilon_alph; assumption.
Qed.
(*									*)
(*  Si on a une transition (e0,epsilon,e) dans d* alors on a 		*)
(*  e0 dans qa et e=g0.							*)
(*									*)

Lemma transition_par_epsilon :
forall (q qa d : Ensf) (g0 e0 e : Elt),
automate q (singleton g0) qa d ->
dans (couple e0 (couple epsilon e)) (fun_d_dstar g0 qa d) ->
dans e0 qa /\ e = g0 :>Elt.