Proposition lub_regular_element_le {H : complete_heyting_algebra} {I : UU} (f : I → regular_element H) (y : regular_element H) (p : ∏ (i : I), f i ≤ y) : lub_regular_element I f ≤ y. Proof. use from_le_regular_element_lattice. refine (pr22 (regular_element_complete_lattice H I f) y _). intros i. use to_le_regular_element_lattice. apply p. 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.