Proposition le_lub_regular_element {H : complete_heyting_algebra} {I : UU} (f : I → regular_element H) (y : regular_element H) (i : I) (p : y ≤ f i) : y ≤ lub_regular_element I f. Proof. refine (cha_le_trans p _). use from_le_regular_element_lattice. exact (pr12 (regular_element_complete_lattice H I f) i). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.OrderTheory.Lattice.Heyting. Require Import UniMath.OrderTheory.Lattice.CompleteHeyting. Require Import UniMath.OrderTheory.Lattice.DerivedLawsCompleteHeyting.