proposition: Lemma AbelianPreCat_homset_assoc_eq2 {X Y : A} (f g : X --> Y) : Abelian_minus_op (AbelianPreCat_homset_zero X Y) (Abelian_op _ _ f g) = Abelian_minus_op (Abelian_minus_op (AbelianPreCat_homset_zero X Y) f) g.
proof: Proof. set (tmp := AbelianPreCat_homset_inv_left' (AbelianPreCat_homset_zero X Y)). apply (maponpaths (λ h : _, Abelian_minus_op h (Abelian_op X Y f g))) in tmp. apply pathsinv0 in tmp. use (pathscomp0 tmp). unfold Abelian_op. rewrite AbelianPreCat_op_eq3. set (tmp2 := AbelianPreCat_homset_zero_left g). unfold Abelian_op in tmp2. rewrite tmp2. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.Monoids UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian