-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M A : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : CommRing A
inst_4 : Algebra R A
x : (↑(PrimeSpectrum.Top R) : Type u)
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
s : (↑(Functor.obj (structurePresheafInModuleCat R M) (op U)) : Type u)
g : R
hxg : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen g) x
igU : basicOpen g ≤ U
f : M
hs :
  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))
            igU.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)
ih :
  Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M))
            (localizationtoStalkₗ R M x) :
          (a : (↑(ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen g)))
                (openToLocalizationₗ R M (basicOpen g) x hxg) :
              (a : (↑((structurePresheafInModuleCat R M).obj (op (basicOpen g))) : Type u)) →
                (↑(ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)) : Type u))
            (const f g (basicOpen g) ⋯) :
          (↑(ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)) : Type u)) :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen g))) (Y :=
            stalk (structurePresheafInModuleCat R M) x) (germ (structurePresheafInModuleCat R M) (basicOpen g) x hxg) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op (basicOpen g))) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op U))
                (CategoryTheory.Functor.map (structurePresheafInModuleCat R M) igU.hom.op) :
              (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
                (↑((structurePresheafInModuleCat R M).obj (op (basicOpen g))) : Type u))
            s :
          (↑(Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen g))) : Type u)) :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
⊢ Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom (openToLocalizationₗ R M U x hxU ≫ localizationtoStalkₗ R M x) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen g))) (Y :=
            stalk (structurePresheafInModuleCat R M) x) (germ (structurePresheafInModuleCat R M) (basicOpen g) x hxg) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op (basicOpen g))) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op U))
                (CategoryTheory.Functor.map (structurePresheafInModuleCat R M) igU.hom.op) :
              (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
                (↑((structurePresheafInModuleCat R M).obj (op (basicOpen g))) : Type u))
            s :
          (↑(Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen g))) : Type u)) :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))

Tactic:
refine congr(localizationtoStalkₗ R M x (openToLocalizationₗ R M _ x hxg $hs)).symm.trans ?_

Premises used: PrimeSpectrum.basicOpen, congrArg, CategoryTheory.ConcreteCategory.hom, _private.0.AlgebraicGeometry.StructureSheaf.localizationtoStalkₗ, id, _private.0.AlgebraicGeometry.StructureSheaf.openToLocalizationₗ, Eq.symm, DFunLike.coe, Eq.trans