Definition ValAdh_un (un:nat -> R) : R -> Prop :=
let D := fun x:R =>  exists n : nat, x = INR n in
let f :=
fun x:R =>
adherence
(fun y:R => (exists p : nat, y = un p /\ x <= INR p) /\ D x) in
intersection_family (mkfamily D f (ValAdh_un_exists un)).

(* Specifications: *)
Lemma ValAdh_un_prop :
forall (un:nat -> R) (x:R), ValAdh un x <-> ValAdh_un un x.
Lemma adherence_P4 :
forall F G:R -> Prop, included F G -> included (adherence F) (adherence G).