theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType := by
  rw [RingHom.ofLocalizationSpan_iff_finite]
  introv R hs H
  -- mirrors the proof of `finite_ofLocalizationSpan`
  letI := f.toAlgebra
  letI := fun r : s => (Localization.awayMap f r).toAlgebra
  have : ∀ r : s,
      IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
    by intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization
  haveI : ∀ r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) :=
    fun r => IsScalarTower.of_algebraMap_eq'
      (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm
  constructor
  replace H := fun r => (H r).1
  choose s₁ s₂ using H
  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s₁ x)
  use s.attach.biUnion sf
  convert (Algebra.adjoin_attach_biUnion (R := R) sf).trans _
  rw [eq_top_iff]
  rintro x -
  apply (⨆ x : s, Algebra.adjoin R (sf x : Set S)).toSubmodule.mem_of_span_eq_top_of_smul_pow_mem
    _ hs _ _
  intro r
  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ :=
    multiple_mem_adjoin_of_mem_localization_adjoin (Submonoid.powers (r : R))
      (Localization.Away (r : R)) (s₁ r : Set (Localization.Away (f r)))
      (algebraMap S (Localization.Away (f r)) x) (by rw [s₂ r]; trivial)
  rw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, ← map_mul] at hn₁
  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ :=
    IsLocalization.lift_mem_adjoin_finsetIntegerMultiple (Submonoid.powers (r : R)) _ (s₁ r) hn₁
  

Proof state:
case intro.mk.intro.intro.mk.intro
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := RingHom.toAlgebra f
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=
  fun r => RingHom.toAlgebra (Localization.awayMap f ↑r)
this✝ :
  ∀ (r : { x // x ∈ s }),
    IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Algebra.adjoin (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ :
  (algebraMap S (Localization.Away (f ↑r)))
      ((algebraMap R S)
          ↑{ val := (fun x => ↑r ^ x) n₁, property := (_ : ∃ y, (fun x => ↑r ^ x) y = (fun x => ↑r ^ x) n₁) } *
        x) ∈
    Algebra.adjoin R ↑(s₁ r)
n₂ : ℕ
hn₂ :
  { val := (fun x => ↑r ^ x) n₂, property := (_ : ∃ y, (fun x => ↑r ^ x) y = (fun x => ↑r ^ x) n₂) } •
      ((algebraMap R S)
          ↑{ val := (fun x => ↑r ^ x) n₁, property := (_ : ∃ y, (fun x => ↑r ^ x) y = (fun x => ↑r ^ x) n₁) } *
        x) ∈
    Algebra.adjoin R
      ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (s₁ r))
⊢ ∃ n, ↑r ^ n • x ∈ Subalgebra.toSubmodule (⨆ x, Algebra.adjoin R ↑(sf x))

Next tactic:
rw [Submonoid.smul_def, ← Algebra.smul_def, smul_smul, Subtype.coe_mk, ← pow_add] at hn₂