Proposition is_scott_continuous_equalizer_pr1 : is_scott_continuous equalizer_dcpo X pr1. Proof. use make_is_scott_continuous. - exact (Equalizer_pr1_monotone X Y f g). - intros D ; cbn. use antisymm_dcpo. + use dcpo_lub_is_least ; cbn. intro i. use less_than_dcpo_lub ; [ exact i | ] ; cbn. apply refl_dcpo. + use dcpo_lub_is_least ; cbn. intro i. use less_than_dcpo_lub ; [ exact i | ] ; cbn. apply refl_dcpo. 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.