proposition: Lemma isExact_Eq {x y z : ob A} (f : x --> y) (g : y --> z) (H : f · g = ZeroArrow (to_Zero A) _ _) : KernelArrow (Abelian.Image f) · g = ZeroArrow (to_Zero A) _ _.
proof: Proof. unfold Abelian.Image. set (fact := factorization1 f). unfold factorization1_monic in fact. cbn in fact. apply (factorization1_is_epi f). rewrite ZeroArrow_comp_right. rewrite assoc. rewrite fact in H. clear fact. exact H. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinDirectSums