(* End of proof of lem_fun_part_inv_eval_prop                              *)

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(* Relation inverse d'une application et applications au cas des bijections*)
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(* On peut definir la relation inverse d'une application                   *)
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Definition fun_recip (f : E) (Af : fun_ f) :=
  subset
    (fun c : E =>
     exists x : E, (exists y : E, In (couple x y) f /\ c = couple y x))
    (cartesien (Img f) (dom f)).

Lemma lem_xyinf_yxinrecipf : forall (f : E) (Af : fun_ f) (x y : E), In (couple x y) f -> In (couple y x) (fun_recip f Af).