(* End of proof of lem_paire_propertie                                     *)

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(* Singleton                                                               *)
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Definition singleton (a : E) := paire a a.

(* Specifications: *)
Lemma lem_singleton_propertie : forall a b : E, singleton a = singleton b <-> a = b.
Lemma lem_in_singleton : forall a x : E, In x (singleton a) <-> x = a.
Lemma lem_x_in_sing_x : forall x : E, In x (singleton x).