-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
x : (↑(PrimeSpectrum.Top R) : Type u)
hx : x ∈ U
obtain :
  ∀ (V : Opens (↑(PrimeSpectrum.Top R) : Type u))
    (hxV :
      (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦ x ∈ unop (op U)) x hx) :
          (↑(PrimeSpectrum.Top R) : Type u)) ∈
        V)
    (iVU : V ⟶ unop (op U)) (f : M) (g : R)
    (hfg :
      ∀ (x : ↥V),
        ∃ (hs : g ∉ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal),
          Eq (α := Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
            ((fun (x : ↥V) ↦
                (↑s : (x : ↥(unop (op U))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
                  ((iVU : (a : ↥V) → ↥(unop (op U))) x))
              x)
            (LocalizedModule.mk f
              (Subtype.mk (p := fun (x_1 : R) ↦ x_1 ∈ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) g
                hs))),
    ∃ (g : R) (_ : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen g) x) (i : basicOpen g ≤ U) (f : M),
      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 g)))
        (const f g (basicOpen g) ⋯)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (CategoryTheory.Functor.map
                (ObjectProperty.FullSubcategory.obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                  (structureSheafInType R M))
                i.hom.op) :
            (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
              (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen g))))
          s)
⊢ ∃ (g : R) (_ : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen g) x) (i : basicOpen g ≤ U) (f : M),
    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 g)))
      (const f g (basicOpen g) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              i.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen g))))
        s)

Tactic:
obtain ⟨V, hxV, iVU, f, g, hfg⟩ := s.2 ⟨x, hx⟩

Premises used: Subtype.mk, Subtype.property