Definition least_upper_bound_in_subposet {P : Poset} {A : Subposet P} {I} {p : I -> A} (x : least_upper_bound (pr1 ∘ p)) (x_A : A x) : least_upper_bound p. Proof. use make_least_upper_bound. { exists x. apply x_A. } apply is_least_upper_bound_in_subposet, least_upper_bound_property. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.