-- Lean 4 proof step from Mathlib.AlgebraicGeometry.AffineScheme (goal)
Goal:
X Y : Scheme
U_1 : X.Opens
hU_1 : IsAffineOpen U_1
f_1 : (↑Γ(X, U_1) : Type u)
f : X ⟶ Y
x : ↥X
U : Y.Opens
hU : IsAffineOpen U
V : X.Opens
hV : IsAffineOpen V
hVU : V ≤ f ⁻¹ᵁ U
hx : x ∈ V
this_1 : Algebra (↑(Y.presheaf.obj (op U)) : Type u)
  (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u) :=
  Y.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)
this_2 :
  IsLocalization.AtPrime (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u)
    (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
this_3 : Algebra (↑(X.presheaf.obj (op V)) : Type u)
  (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u) :=
  X.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)
this :
  IsLocalization.AtPrime (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u)
    (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
a : (↑Γ(Y, U) : Type u)
⊢ Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((RingHom.comp (β := (↑(Y.presheaf.stalk (f x)) : Type u))
            (CommRingCat.Hom.hom
              (CategoryStruct.comp (Y :=
                CommRingCat.of
                  (Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal))
                (CategoryStruct.comp (Y :=
                  CommRingCat.of
                    (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal))
                  (Iso.hom
                    (RingEquiv.toCommRingCatIso
                      (AlgEquiv.toRingEquiv
                        (IsLocalization.algEquiv
                          (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal.primeCompl
                          (↑(Y.presheaf.stalk (f x)) : Type u)
                          (Localization.AtPrime
                            (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal)))))
                  (CommRingCat.ofHom
                    (Localization.localRingHom
                      (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
                      (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
                      (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) ⋯)))
                (Iso.inv
                  (RingEquiv.toCommRingCatIso
                    (AlgEquiv.toRingEquiv
                      (IsLocalization.algEquiv
                        (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl
                        (↑(X.presheaf.stalk x) : Type u)
                        (Localization.AtPrime
                          (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal)))))))
            (algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u)) :
          (a : (↑Γ(Y, U) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        a :
      (↑(X.presheaf.stalk x) : Type u))
    (((CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x)).comp
            (algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u)) :
          (a : (↑Γ(Y, U) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        a :
      (↑(X.presheaf.stalk x) : Type u))
have refine_3 : Eq (α := (↑(X.presheaf.stalk x) : Type u))
  ((IsLocalization.map (R := (↑Γ(X, V) : Type u)) (M :=
          (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl) (T :=
          (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl)
          (↑(X.presheaf.stalk x) : Type u) (RingHom.id (↑Γ(X, V) : Type u)) sorry :
        (a : Localization.AtPrime (hV.primeIdealOf ⟨x, hx⟩).asIdeal) → (↑(X.presheaf.stalk x) : Type u))
      ((Localization.localRingHom (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal
              (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
              (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) sorry :
            (a : Localization.AtPrime (hU.primeIdealOf ⟨f x, sorry⟩).asIdeal) →
              Localization.AtPrime (hV.primeIdealOf ⟨x, hx⟩).asIdeal)
          ((IsLocalization.map (R := (↑Γ(Y, U) : Type u)) (M :=
                (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal.primeCompl) (T :=
                (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal.primeCompl)
                (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) sorry)).asIdeal)
                (RingHom.id (↑Γ(Y, U) : Type u)) sorry :
              (a : (↑(Y.presheaf.stalk (f x)) : Type u)) → Localization.AtPrime (hU.primeIdealOf ⟨f x, sorry⟩).asIdeal)
            ((algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u) :
                (a : (↑Γ(Y, U) : Type u)) → (↑(Y.presheaf.stalk (f x)) : Type u))
              a)) :
        Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal) :
    (↑(X.presheaf.stalk x) : Type u))
  ((CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x) :
        (a : (↑(Y.presheaf.stalk (f x)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
      ((algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u) :
            (a : (↑Γ(Y, U) : Type u)) → (↑(Y.presheaf.stalk (f x)) : Type u))
          a :
        (↑(Y.presheaf.stalk (f x)) : Type u)) :
    (↑(X.presheaf.stalk x) : Type u))
Tactic: dsimp [← AlgEquiv.symm_toRingEquiv]