Proposition square_to_vertical_cell_comp (HB : vertically_saturated B) {x y : B} {v₁ v₂ v₃ : x -|-> y} (s₁ : square_double_bicat (id_h x) (id_h y) v₁ v₂) (s₂ : square_double_bicat (id_h x) (id_h y) v₂ v₃) : square_to_vertical_cell HB (comp_ver_globular_square s₁ s₂) = square_to_vertical_cell HB s₁ • square_to_vertical_cell HB s₂. Proof. use (invmaponpathsweq (vertical_cell_to_square_weq HB v₁ v₃)) ; cbn. rewrite vertical_cell_to_square_comp. rewrite !square_to_vertical_cell_to_square. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws.