Section given_an_element.
Variable A : Ens.
Variable X : Ens.
Variable k : IN X A.
Hypothesis m : has_leq_0_elements A.
Definition given_an_element_Out : False := m X k.
End given_an_element.
Proof given_an_element_Out.
(**** the next statement was already proved above with different notations but we can just redefine it, coq takes care of the change of notations ! *****)

Theorem Singletons_have_geq_1_elements :
forall X : Ens, has_geq_1_elements (Singleton X).