Definition family_closed_set (f:family) : Prop :=
  forall x:R, closed_set (f x).Definition intersection_vide_in (D:R -> Prop) (f:family) : Prop :=
  forall x:R,
    (ind f x -> included (f x) D) /\
    ~ (exists y : R, intersection_family f y).Definition intersection_vide_finite_in (D:R -> Prop)
  (f:family) : Prop := intersection_vide_in D f /\ family_finite f.

(* Specifications: *)
Lemma compact_P6 : forall X:R -> Prop, compact X -> (exists z : R, X z) -> forall g:family, family_closed_set g -> intersection_vide_in X g -> exists D : R -> Prop, intersection_vide_finite_in X (subfamily g D).
Theorem Bolzano_Weierstrass : forall (un:nat -> R) (X:R -> Prop), compact X -> (forall n:nat, X (un n)) -> exists l : R, ValAdh un l.