Proposition is_sharp_is_strongly_maximal {X : dcpo} {x : X} (Hx : is_strongly_maximal x) : is_sharp x. Proof. intros u v p. specialize (Hx u v p). revert Hx. use factor_through_squash. { apply propproperty. } intro H. induction H as [ q | q ]. - exact (hinhpr (inl q)). - refine (hinhpr (inr _)). apply is_hausdorff_separated_not_le. exact q. 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.Core.ScottTopology. Require Import UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Require Import UniMath.OrderTheory.DCPOs.Basis.Basis. Require Import UniMath.OrderTheory.DCPOs.Elements.Sharp.