-- Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n₁ : ℕ
x₁ : Ext.{w, v, u} X S.X₁ n₁
hx₁ : x₁.comp (mk₀ S.f) ⋯ = 0
n₀ : ℕ
hn₀ : n₀ + 1 = n₁
this :
  (ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀))
      (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯)) ⋯).Exact
⊢ ∀
    (x₂ :
      (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
              (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯)) ⋯).X₂ :
        Type w)),
    Eq (α :=
        (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                ⋯).X₃ :
          Type w))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
                AddCommGrpCat.carrier)
                (ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                    (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                    ⋯).g :
              (a :
                  (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
                          X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₀),
                          g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.X₂ :
                    Type w)) →
                (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
                        X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₀),
                        g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.X₃ :
                  Type w))
            x₂ :
          (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                  (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                  ⋯).X₃ :
            Type w))
        (0 :
          (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                  (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                  ⋯).X₃ :
            Type w)) →
      ∃ (x₁ :
        (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                ⋯).X₁ :
          Type w)),
        (ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
                AddCommGrpCat.carrier)
                (ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₃ n₀))
                    (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₀)) (AddCommGrpCat.ofHom (postcomp (mk₀ S.f) X ⋯))
                    ⋯).f :
              (a :
                  (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
                          X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₀),
                          g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.X₁ :
                    Type w)) →
                (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
                        X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₀),
                        g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.X₂ :
                  Type w))
            x₁ =
          x₂

Tactic:
rw [ShortComplex.ab_exact_iff] at this

Premises used: CategoryTheory.ShortComplex.ab_exact_iff