Proposition way_below_prod_weq (xy₁ xy₂ : prod_dcpo X Y) : xy₁ ≪ xy₂ ≃ ((pr1 xy₁ ≪ pr1 xy₂) ∧ (pr2 xy₁ ≪ pr2 xy₂)). Proof. use weqimplimpl. - intros p. split. + exact (way_below_prod_pr1 p). + exact (way_below_prod_pr2 p). - intros p. exact (to_way_below_prod (pr1 p) (pr2 p)). - apply propproperty. - apply propproperty. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.PointedPosets. Require Import UniMath.OrderTheory.Posets.MonotoneFunctions. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.ScottContinuous. Require Import UniMath.OrderTheory.DCPOs.Core.ScottTopology. Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Basis.Basis. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Require Import UniMath.OrderTheory.DCPOs.Elements.Maximal.