-- Module: Mathlib.Algebra.Homology.HomologicalComplex
Goal:
V : Type u
inst : Category.{v, u} V
inst_1 : HasZeroMorphisms.{v, u} V
X₀ X₁ X₂ : V
d₀ : X₁ ⟶ X₀
d₁ : X₂ ⟶ X₁
s : d₁ ≫ d₀ = 0
succ : (S : ShortComplex.{v, u} V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0
n : ℕ
⊢ PSigma.fst (β :=
      fun
        (d₂ :
          PSigma.fst (β := fun (X₃ : V) ↦
              (d₂ : X₃ ⟶ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).X₁) ×' d₂ ≫ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).f = 0)
              (succ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n)) ⟶
            (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).X₁) ↦
      d₂ ≫ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).f = 0)
      (PSigma.snd (β := fun (X₃ : V) ↦
        (d₂ : X₃ ⟶ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).X₁) ×' d₂ ≫ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n).f = 0)
        (succ (mkAux X₀ X₁ X₂ d₀ d₁ s succ n))) =
    eqToHom ⋯ ≫
      PSigma.fst (β :=
          fun
            (d₂ :
              PSigma.fst (β := fun (X₃ : V) ↦
                  (d₂ :
                    X₃ ⟶
                      (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                          (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                          ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).X₁) ×'
                    d₂ ≫
                        (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                            (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                            ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).f =
                      0)
                  (succ
                    (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                      (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                      ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯)) ⟶
                (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                    (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                    ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).X₁) ↦
          d₂ ≫
              (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1))
                  ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).f =
            0)
          (PSigma.snd (β := fun (X₃ : V) ↦
            (d₂ :
              X₃ ⟶
                (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                    (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                    ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).X₁) ×'
              d₂ ≫
                  (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ :=
                      (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1))
                      ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯).f =
                0)
            (succ
              (ShortComplex.mk (X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 2)) (X₂ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1))
                ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 2) (n + 1)) ((mk X₀ X₁ X₂ d₀ d₁ s succ).d (n + 1) n) ⋯))) ≫
        eqToHom ⋯

Tactic:
have eq := mk_congr_succ_d₂ succ
    (mkAux_eq_shortComplex_mk_d_comp_d X₀ X₁ X₂ d₀ d₁ s succ n)