Lemma bracket_Thm15_ok (Z: Ptd)(f : Ptd ⟦ Z, ptd_from_alg InitAlg ⟧): bracket_property_parts (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) _ f ⦃f⦄. Proof. split. + exact (bracket_Thm15_ok_part1 Z f). + exact (bracket_Thm15_ok_part2 Z f). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Notation.