-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
h_aJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
a₁ : R :=
  choose (p := fun (x : R) ↦
    ∃ (aI : Ideal R),
      x ≠ 0 ∧
        I =
          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
            (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
            ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
    ⋯
J₁ : Ideal R :=
  choose (p := fun (x : Ideal R) ↦
    choose (p := fun (x : R) ↦
          ∃ (aI : Ideal R),
            x ≠ 0 ∧
              I =
                HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                  FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                  (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                  ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
          ⋯ ≠
        0 ∧
      I =
        HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
          (spanSingleton R⁰
              ((algebraMap R K : (a : R) → K)
                  (choose (p := fun (x : R) ↦
                    ∃ (aI : Ideal R),
                      x ≠ 0 ∧
                        I =
                          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                    ⋯))⁻¹ :
            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
          ((↑x : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
    ⋯
⊢ count K v I =
    HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ)
have h_a₁J₁ : I =
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a₁)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    ((↑J₁ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
Tactic: have h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁ :=
    (choose_spec (choose_spec (exists_eq_spanSingleton_mul I))).2