proposition: Lemma commutative_symmetric_braiding_after_4_copies (m : commutative_comonoid M) : δ_{m} · (δ_{m} #⊗ δ_{m} · (α^{M}_{ m, m, m ⊗_{ M} m} · (m ⊗^{M}_{l} (αinv^{M}_{ m, m, m} · (sym_mon_braiding M m m ⊗^{M}_{r} m · α^{M}_{ m, m, m})) · αinv^{M}_{m, m, m ⊗_{ M} m}))) = δ_{m} · δ_{m} #⊗ δ_{m}.
proof: Proof. rewrite ! assoc. etrans. { apply maponpaths_2. rewrite ! (bifunctor_leftcomp M). rewrite ! assoc. apply maponpaths_2. exact (commutative_symmetric_braiding_after_4_copies' _). } etrans. { apply maponpaths_2. rewrite assoc'. apply maponpaths. rewrite <- (bifunctor_leftcomp M). apply maponpaths. apply (monoidal_associatorisolaw M m m m). } rewrite (bifunctor_leftid M). rewrite id_right. etrans. { rewrite assoc'. apply maponpaths. apply (monoidal_associatorisolaw M m m _). } apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Projection UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Examples.Sigma UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Examples.DiagonalFunctor UniMath.CategoryTheory.Monoidal.Examples.ConstantFunctor UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.CategoryTheory.Categories.Dialgebras