-- Module: Mathlib.Algebra.Module.FinitePresentation
Goal:
R : Type u_3
M : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
S : Submonoid R
M' : Type u_1
inst_3 : AddCommGroup M'
inst_4 : Module R M'
f : M →ₗ[R] M'
inst_5 : IsLocalizedModule S f
inst_6 : Module.Finite R M
inst_7 : Module.FinitePresentation R M'
this : IsLocalizedModule.{u_3, u_1, u_1} (M := M') S LinearMap.id
r : R
hrp : r ∈ S
H :
  ∀ (t : R),
    r ∣ t →
      Function.Bijective (α := LocalizedModule (Submonoid.powers t) M) (β := LocalizedModule (Submonoid.powers t) M')
        (⇑((LocalizedModule.map (Submonoid.powers t) :
                  (a : M →ₗ[R] M') →
                    LocalizedModule (Submonoid.powers t) M →ₗ[Localization (Submonoid.powers t)]
                      LocalizedModule (Submonoid.powers t) M')
                f :
              LinearMap (R := Localization (Submonoid.powers t)) (RingHom.id (Localization (Submonoid.powers t)))
                (LocalizedModule (Submonoid.powers t) M) (LocalizedModule (Submonoid.powers t) M')) :
          (a : LocalizedModule (Submonoid.powers t) M) → LocalizedModule (Submonoid.powers t) M')
hrp' : Submonoid.powers r ≤ S
y : M'
x : LocalizedModule (Submonoid.powers r) M
hx :
  Eq (α := LocalizedModule (Submonoid.powers r) M')
    ((((LocalizedModule.map (Submonoid.powers r) :
                (a : M →ₗ[R] M') →
                  LocalizedModule (Submonoid.powers r) M →ₗ[Localization (Submonoid.powers r)]
                    LocalizedModule (Submonoid.powers r) M')
              f :
            LinearMap (R := Localization (Submonoid.powers r)) (RingHom.id (Localization (Submonoid.powers r)))
              (LocalizedModule (Submonoid.powers r) M) (LocalizedModule (Submonoid.powers r) M')) :
          (a : LocalizedModule (Submonoid.powers r) M) → LocalizedModule (Submonoid.powers r) M')
        x :
      LocalizedModule (Submonoid.powers r) M')
    ((LocalizedModule.mkLinearMap (Submonoid.powers r) M' : (a : M') → LocalizedModule (Submonoid.powers r) M') y :
      LocalizedModule (Submonoid.powers r) M')
refine_1 :
  ∀ (x : M) (n : ℕ)
    (hx :
      Eq (α := LocalizedModule (Submonoid.powers r) M')
        ((((LocalizedModule.map (Submonoid.powers r) :
                    (a : M →ₗ[R] M') →
                      LocalizedModule (Submonoid.powers r) M →ₗ[Localization (Submonoid.powers r)]
                        LocalizedModule (Submonoid.powers r) M')
                  f :
                LinearMap (R := Localization (Submonoid.powers r)) (RingHom.id (Localization (Submonoid.powers r)))
                  (LocalizedModule (Submonoid.powers r) M) (LocalizedModule (Submonoid.powers r) M')) :
              (a : LocalizedModule (Submonoid.powers r) M) → LocalizedModule (Submonoid.powers r) M')
            (Function.uncurry (mk' (LocalizedModule.mkLinearMap (Submonoid.powers r) M))
              (x,
                Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers r) x)
                  ((fun (x : ℕ) ↦ r ^ x) n) ⋯)) :
          LocalizedModule (Submonoid.powers r) M')
        ((LocalizedModule.mkLinearMap (Submonoid.powers r) M' : (a : M') → LocalizedModule (Submonoid.powers r) M') y :
          LocalizedModule (Submonoid.powers r) M')),
    ∃ (x : Prod.{u_4, u_3} M ↥(Submonoid.powers r)), x.2 • y = (f : (a : M) → M') x.1
⊢ ∃ (x : Prod.{u_4, u_3} M ↥(Submonoid.powers r)), x.2 • y = (f : (a : M) → M') x.1

Tactic:
obtain ⟨⟨x, ⟨_, n, rfl⟩⟩, rfl⟩ := IsLocalizedModule.mk'_surjective
      (.powers r) (LocalizedModule.mkLinearMap _ _) x

Premises used: IsLocalizedModule.mk'_surjective, Submonoid.powers, LocalizedModule.mkLinearMap, rfl