Proposition symm_is_hausdorff_separated {x y : X} (H : is_hausdorff_separated x y) : is_hausdorff_separated y x. Proof. revert H. use factor_through_squash. { apply propproperty. } intros H. induction H as [ P₁ [ P₂ [ Px [ Py H ]]]]. refine (hinhpr (P₂ ,, P₁ ,, Py ,, Px ,, _)). apply symm_are_disjoint. exact H. 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.