(* Proof of lem_not_vide_in_vide                                           *)
absurd (Vide = Vide); [ unfold not in |- *; intros | reflexivity ].
generalize (lem_x_noteq_vide Vide Vide H); intros.
tauto.
Qed.
(* End of proof of lem_not_vide_in_vide                                    *)
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(* Intersection de deux ensembles                                          *)
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Definition def_F_inter (s x : E) := In x s.
Definition inter (a b : E) := subset (def_F_inter b) a.

(* Specifications: *)
Lemma lem_inter_propertie :
forall a b x : E, In x (inter a b) <-> In x a /\ In x b.