Proposition hset_struct_smash_prod_lassociator_mor_alt_eq3 {X Y Z : category_of_hset_struct P} (xy₁ xy₂ : pr11 (X ∧* Y)) : hset_struct_smash_prod_lassociator_mor_alt_fun xy₁ (hset_struct_point (pr12 P) (pr2 Z)) = hset_struct_smash_prod_lassociator_mor_alt_fun xy₂ (hset_struct_point (pr12 P) (pr2 Z)). Proof. revert xy₁. use setquotunivprop'. { intro. apply setproperty. } intros xy₁. induction xy₁ as [ x₁ y₁ ]. revert xy₂. use setquotunivprop'. { intro. apply setproperty. } intros xy₂. induction xy₂ as [ x₂ y₂ ]. use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split ; use inr ; refine (_ @ !(hset_struct_with_smash_closed_point_smash _ _)). - use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split. + use inr. apply idpath. + use inr. apply idpath. - use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split. + use inr. apply idpath. + use inr. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructuresSmashProduct. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Terminal.