Proposition is_scott_continuous_on_lub {X Y : hSet} {DX : dcpo_struct X} {DY : dcpo_struct Y} {f : X → Y} (Hf : is_scott_continuous DX DY f) {I : UU} (D : I → X) (HD : is_directed DX D) : f (dcpo_struct_lub DX D HD) = dcpo_struct_lub DY (λ i, f (D i)) (is_directed_comp f (pr1 Hf) HD). Proof. use (eq_lub DY (λ i, f(D i))). - use (is_scott_continuous_lub Hf HD). apply dcpo_struct_lub. - apply dcpo_struct_lub. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.MonotoneFunctions. Require Import UniMath.OrderTheory.Posets.PointedPosets. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.ScottTopology. Require Import UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness.