proposition: Lemma AbelianPreCat_op_eq3 {X Y : A} (a b c d : X --> Y) : Abelian_minus_op (Abelian_minus_op a b) (Abelian_minus_op c d) = Abelian_minus_op (Abelian_minus_op a c) (Abelian_minus_op b d).
proof: Proof. unfold Abelian_minus_op at 1. set (tmp := AbelianPreCat_op_eq1 a c b d). unfold Abelian_mor_to_BinProd in tmp. rewrite <- tmp. set (com := Abelian_op_eq (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))). unfold Abelian_minus_op at 1. rewrite <- assoc. rewrite com. rewrite assoc. set (tmp1 := AbelianPreCat_op_eq2 a c b d). unfold Abelian_mor_to_BinProd, Abelian_mor_from_to_BinProd in tmp1. rewrite tmp1. apply cancel_postcomposition. unfold Abelian_minus_op. 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