Proposition compact_approximating_family_lub (x : X) : ⨆ (compact_approximating_family x) = x. Proof. use antisymm_dcpo. - use dcpo_lub_is_least. intro i. exact (pr122 (AX x) i). - use (is_least_upperbound_is_least (pr22 (AX x))). apply is_least_upperbound_is_upperbound. apply is_least_upperbound_dcpo_lub. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous.