Proposition map_from_basis_monotone (x₁ x₂ : X) (p : x₁ ⊑ x₂) : map_from_basis x₁ ⊑ map_from_basis x₂. Proof. unfold map_from_basis. use dcpo_lub_is_least ; cbn. intro i. use less_than_dcpo_lub. - refine (pr1 i ,, _). exact (trans_way_below_le (pr2 i) p). - cbn. apply refl_dcpo. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets. 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.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Section ScottContinuousFromBasis. End ScottContinuousFromBasis.