-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (hypothesis)
Goal:
R S : CommRingCat
p : PrimeSpectrum (↑R : Type u)
inst : Algebra (↑R : Type u) (↑S : Type u)
y :
  (↑(TopCat.Presheaf.stalk
        (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
          (TopCat.Presheaf.pushforward CommRingCat
            (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
          (ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
            (structureSheaf (↑S : Type u))))
        p) :
    Type u)
U : TopologicalSpace.Opens (↑(Spec.topObj (CommRingCat.of (↑R : Type u))) : Type u)
hp : p ∈ U
s :
  ToType (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier)
    (Functor.obj
      (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
        (TopCat.Presheaf.pushforward CommRingCat
          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
        (ObjectProperty.FullSubcategory.obj (P :=
          Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
          (structureSheaf (↑S : Type u))))
      (op U))
e :
  (ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (X :=
          Functor.obj
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            (op U))
          (Y :=
          TopCat.Presheaf.stalk
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            p)
          (TopCat.Presheaf.germ
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            U p hp) :
        (a :
            (↑(((TopCat.Presheaf.pushforward CommRingCat
                          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                      (structureSheaf (↑S : Type u)).obj).obj
                  (op U)) :
              Type u)) →
          (↑(((TopCat.Presheaf.pushforward CommRingCat
                        (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                    (structureSheaf (↑S : Type u)).obj).stalk
                p) :
            Type u))
      s =
    y
r : (↑(CommRingCat.of (↑R : Type u)) : Type u)
hpr :
  Membership.mem (γ := TopologicalSpace.Opens (PrimeSpectrum (↑(CommRingCat.of (↑R : Type u)) : Type u)))
    (PrimeSpectrum.basicOpen r) p
hrU : PrimeSpectrum.basicOpen r ≤ U
⊢ ∃ (x : (↑S : Type u) × ↥p.asIdeal.primeCompl),
    x.2 • y =
      (toPushforwardStalkAlgHom R S p :
          (a : (↑S : Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat
                          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                      (structureSheaf (↑S : Type u)).obj).stalk
                  p) :
              Type u))
        x.1
have e₁ : (ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier)
        (X :=
        Functor.obj
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          (op (PrimeSpectrum.basicOpen r)))
        (Y :=
        TopCat.Presheaf.stalk
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          p)
        (TopCat.Presheaf.germ
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          (PrimeSpectrum.basicOpen r) p hpr) :
      (a :
          (↑(((TopCat.Presheaf.pushforward CommRingCat
                        (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                    (structureSheaf (↑S : Type u)).obj).obj
                (op (PrimeSpectrum.basicOpen r))) :
            Type u)) →
        (↑(((TopCat.Presheaf.pushforward CommRingCat
                      (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                  (structureSheaf (↑S : Type u)).obj).stalk
              p) :
          Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            Functor.obj
              (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
                (TopCat.Presheaf.pushforward CommRingCat
                  (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                  (structureSheaf (↑S : Type u))))
              (op U))
            (CategoryTheory.Functor.map
              (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
                (TopCat.Presheaf.pushforward CommRingCat
                  (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                  (structureSheaf (↑S : Type u))))
              (homOfLE hrU).op) :
          (a :
              (↑(((TopCat.Presheaf.pushforward CommRingCat
                            (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                        (structureSheaf (↑S : Type u)).obj).obj
                    (op U)) :
                Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat
                          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                      (structureSheaf (↑S : Type u)).obj).obj
                  (op (PrimeSpectrum.basicOpen r))) :
              Type u))
        s :
      (↑(Functor.obj
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            (op (PrimeSpectrum.basicOpen r))) :
        Type u)) =
  y
Tactic: replace e :=
    ((Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).germ_res_apply
      (homOfLE hrU) p hpr _).trans e
Cleared hypotheses: e