Proposition approximating_family_lub (x : X) : ⨆ (approximating_family x) = x. Proof. use antisymm_dcpo. - use dcpo_lub_is_least. intro i. apply way_below_to_le. exact (approximating_family_way_below x i). - use (is_least_upperbound_is_least (pr22 (CX 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.