Definition quotient_lower_set : PartialOrder (setquotinset downward_closed_to_eqrel). Proof. use make_PartialOrder. - exact quotient_lower_set_order. - refine ((_ ,, _) ,, _). + exact istrans_quotient_lower_set_order. + exact isrefl_quotient_lower_set_order. + exact isantisymm_quotient_lower_set_order. Defined.
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.