Definition is_least_upper_bound_in_subposet {P : Poset} {A : Subposet P} {I} {p : I -> A} {x : A} (x_lub : is_least_upper_bound (subposet_incl ∘ p) (subposet_incl x)) : is_least_upper_bound p x. Proof. split. - apply is_upper_bound_in_subposet, x_lub. - intros [x' x'_ub]. apply (least_upper_bound_univ (make_least_upper_bound _ x_lub)). apply is_upper_bound_in_subposet; assumption. 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.