Definition is_least_upper_bound_if_pointwise {X:UU} {P : X -> Poset} {I} {f : I -> forall_Poset P} {g : forall_Poset P} : is_least_upper_bound_pointwise f g -> is_least_upper_bound f g. Proof. intro pointwise_lub. set (g_pwlub := fun x => make_least_upper_bound _ (pointwise_lub x)). split. - intros i x. apply (least_upper_bound_is_upper_bound (g_pwlub x)). - intros [g' g'_ub] x. apply (least_upper_bound_univ (g_pwlub x)). apply isupperbound_if_pointwise, g'_ub. 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.