Lemma SE2NW_and_SW2NE_is_id {X : hSet} (b : interlaced_prebilattice X) (x y : X) (p1 : rel_same_SW2NE b x y) (p2 : rel_same_SE2NW b x y) : x = y. Proof. set (p := p1 @ !p2). use (isantisymm_Lle (tlattice b)). - use (istrans_Lle (tlattice _) x (join b x y) y (Lmax_le_l _ _ _)). rewrite p. use Lmin_le_r. - use (istrans_Lle (tlattice _) y (join b x y) x (Lmax_le_r _ _ _)). rewrite p. use Lmin_le_l. Defined.
Require Import UniMath.OrderTheory.Prebilattice.Prebilattice. Require Import UniMath.OrderTheory.Prebilattice.Interlaced. Require Import UniMath.OrderTheory.Prebilattice.PrebilatticeDisplayedCat. Require Import UniMath.MoreFoundations.Equivalences. Require Import UniMath.Foundations.Sets. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.CategoryTheory.Core.Prelude.