-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
t : LocalizedModule.Away f M
a : M
b : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk a b) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f))) t :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
h :
  ∀ (c : M) (d : ↥(Submonoid.powers f))
    (h_eq :
      Eq (α :=
        Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
            (structureSheafInType R M))
          (op (basicOpen f)))
        ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
            (LocalizedModule.mk a b) :
          Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            (op (basicOpen f)))
        ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
            (LocalizedModule.mk c d) :
          Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            (op (basicOpen f)))),
    Eq (α := LocalizedModule.Away f M) (LocalizedModule.mk a b) (LocalizedModule.mk c d)
⊢ LocalizedModule.mk a b = t

Tactic:
induction t using LocalizedModule.induction_on with | h c d =>
  suffices f ∈ ((⊥ : Submodule R M).colon {d • a - b • c}).radical by
    rw [LocalizedModule.mk_eq]
    obtain ⟨n, hn⟩ := this
    exact ⟨⟨f ^ n, n, rfl⟩, by simpa [sub_eq_zero, smul_sub] using Submodule.mem_colon.mp hn _ rfl⟩
  simp only [toBasicOpenₗ_mk] at h_eq
  rw [← PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.mem_vanishingIdeal]
  intro p hfp
  contrapose hfp
  obtain ⟨u, hu⟩ := LocalizedModule.mk_eq.mp congr(($h_eq).1 ⟨p, hfp⟩)
  rw [PrimeSpectrum.mem_zeroLocus, Set.not_subset]
  exact ⟨u.1, by simpa [sub_eq_zero, smul_sub], u.2⟩

Premises used: PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.mem_zeroLocus, Set.not_subset, PrimeSpectrum.mem_vanishingIdeal, _private.0.AlgebraicGeometry.StructureSheaf.toBasicOpenₗ_mk