Proposition hset_struct_smash_prod_rassociator_mor_alt_eq1 {X Y Z : ∁} (yz₁ yz₂ : pr11 (Y ∧* Z)) : hset_struct_smash_prod_rassociator_mor_alt_fun (hset_struct_point (pr12 P) (pr2 X)) yz₁ = hset_struct_smash_prod_rassociator_mor_alt_fun (hset_struct_point (pr12 P) (pr2 X)) yz₂. Proof. revert yz₁. use setquotunivprop'. { intro. apply setproperty. } intros yz₁. induction yz₁ as [ y₁ z₁ ]. revert yz₂. use setquotunivprop'. { intro. apply setproperty. } intros yz₂. induction yz₂ as [ y₂ z₂ ]. use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split ; use inl ; refine (_ @ !(hset_struct_with_smash_closed_point_smash _ _)). - use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split. + use inl. apply idpath. + use inr. apply idpath. - use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. split. + use inl. 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.