Definition least_upper_bound_subtype_univ {A : hsubtype P} (x : least_upper_bound (pr1carrier A)) (x' : P) : x ≤ x' <-> (forall y, A y -> y ≤ x'). Proof. split. - intros H y A_y. refine (pr1 (least_upper_bound_univ x x') H (_,,_)); assumption. - intros H. apply least_upper_bound_univ. intros [y A_y]; apply H; assumption. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.