Definition interlaced_prebilattice_eq_product {X : hSet} (b : interlaced_prebilattice X) : ∑ (X1 X2 : hSet) (l1 : lattice X1) (l2 : lattice X2) (b' : prod_prebilattice X1 X2 l1 l2), (X,, (interlaced_prebilattice_to_prebilattice b)) = (prod_prebilattice_carrier X1 X2,, (prod_prebilattice_to_prebilattice b')). Proof. set (interlacedToProd := interlaced_prebilattice_to_prod b). exists (pr1 interlacedToProd). exists (pr12 interlacedToProd). exists (pr122 interlacedToProd). exists (pr1 (pr222 interlacedToProd)). exists (pr2 (pr222 interlacedToProd)). exact (isotoid prebilattice_disp_category is_univalent_prebilattice_disp_category (iso_interlaced_product b)). 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.