-- 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
n₀ : ℕ
x₃ : Ext.{w, v, u} X S.X₃ n₀
n₁ : ℕ
hn₁ : n₀ + 1 = n₁
hx₃ : x₃.comp hS.extClass hn₁ = 0
this :
  ∀
    (x₂ :
      (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₂ n₀))
              (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁)) ⋯).X₂ :
        Type w)),
    Eq (α :=
        (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₂ n₀))
                (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                ⋯).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.X₂ n₀))
                    (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                    ⋯).g :
              (a :
                  (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                          X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                          g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.X₂ :
                    Type w)) →
                (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                        X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                        g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.X₃ :
                  Type w))
            x₂ :
          (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₂ n₀))
                  (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                  ⋯).X₃ :
            Type w))
        (0 :
          (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₂ n₀))
                  (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                  ⋯).X₃ :
            Type w)) →
      ∃ (x₁ :
        (↑(ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} X S.X₂ n₀))
                (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                ⋯).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.X₂ n₀))
                    (AddCommGrpCat.ofHom (postcomp (mk₀ S.g) X ⋯)) (AddCommGrpCat.ofHom (postcomp hS.extClass X hn₁))
                    ⋯).f :
              (a :
                  (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                          X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                          g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.X₁ :
                    Type w)) →
                (↑{ X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                        X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                        g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.X₂ :
                  Type w))
            x₁ =
          x₂
⊢ ∃ (x₂ : Ext.{w, v, u} X S.X₂ n₀), x₂.comp (mk₀ S.g) ⋯ = x₃

Tactic:
this x₃ hx₃

Premises used: CategoryTheory.Localization.SmallHom, CategoryTheory.Abelian.Ext.comp, CategoryTheory.Localization.SmallHom.comp, CategoryTheory.Abelian.Ext, CategoryTheory.Localization.SmallShiftedHom.comp, CategoryTheory.Localization.SmallShiftedHom