Proposition hset_struct_smash_prod_lassociator_mor_alt_eq2 {X Y Z : category_of_hset_struct P} (xy : pr11 (X ∧* Y)) (z : pr11 Z) : hset_struct_smash_prod_lassociator_mor_alt_fun xy (hset_struct_point (pr12 P) (pr2 Z)) = hset_struct_smash_prod_lassociator_mor_alt_fun (hset_struct_point (pr12 P) (pr2 (X ∧* Y))) z. Proof. revert xy. use setquotunivprop'. { intro. apply setproperty. } intros xy. induction xy as [ x y ]. refine (!_). etrans. { apply maponpaths_2. apply 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. 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. 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.