-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : LocallyRingedSpace
f : X ⟶ Z
g : Y ⟶ Z
H : IsOpenImmersion f
s : PullbackCone f g
x : (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u_1)
⊢ IsLocalHom (R :=
    (↑((pullbackConeOfLeft f g).pt.presheaf.stalk
          (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  TopCat.carrier) (X := (↑s.pt.toPresheafedSpace : TopCat)) (Y :=
                  (↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat))
                  (PresheafedSpace.Hom.base
                      (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift f.toHom g.toHom
                        (PullbackCone.mk s.fst.toHom s.snd.toHom ⋯)) :
                    (↑s.pt.toPresheafedSpace : TopCat) ⟶ (↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat)) :
                C((↑(↑s.pt.toPresheafedSpace : TopCat) : Type u_1),
                  (↑(↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat) : Type u_1))) :
              (a : (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u_1)) →
                (↑(↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat) : Type u_1))
            x)) :
      Type u_1))
    (S := (↑(s.pt.presheaf.stalk x) : Type u_1)) (F :=
    (↑((pullbackConeOfLeft f g).pt.presheaf.stalk
            (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    TopCat.carrier) (X := (↑s.pt.toPresheafedSpace : TopCat)) (Y :=
                    (↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat))
                    (PresheafedSpace.Hom.base
                        (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift f.toHom g.toHom
                          (PullbackCone.mk s.fst.toHom s.snd.toHom ⋯)) :
                      (↑s.pt.toPresheafedSpace : TopCat) ⟶ (↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat)) :
                  C((↑(↑s.pt.toPresheafedSpace : TopCat) : Type u_1),
                    (↑(↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat) : Type u_1))) :
                (a : (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u_1)) →
                  (↑(↑(pullbackConeOfLeft f g).pt.toPresheafedSpace : TopCat) : Type u_1))
              x)) :
        Type u_1) →+*
      (↑(s.pt.presheaf.stalk x) : Type u_1))
    (CommRingCat.Hom.hom
      (PresheafedSpace.Hom.stalkMap
        (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift f.toHom g.toHom
          (PullbackCone.mk s.fst.toHom s.snd.toHom ⋯))
        x))
have this : Eq (α :=
  Y.presheaf.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (X :=
            (↑(Cone.pt (F := cospan f.toHom g.toHom) (PullbackCone.mk s.fst.toHom s.snd.toHom sorry)) : TopCat))
            (PresheafedSpace.Hom.base
              (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift f.toHom g.toHom
                  (PullbackCone.mk s.fst.toHom s.snd.toHom sorry) ≫
                (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeft f.toHom g.toHom).snd)) :
          (a : (↑(↑(PullbackCone.mk s.fst.toHom s.snd.toHom sorry).pt : TopCat) : Type u_1)) →
            (↑(↑Y.toPresheafedSpace : TopCat) : Type u_1))
        x) ⟶
    TopCat.Presheaf.stalk
      (PresheafedSpace.presheaf (Cone.pt (F := cospan f.toHom g.toHom) (PullbackCone.mk s.fst.toHom s.snd.toHom sorry))) x)
  (PresheafedSpace.Hom.stalkMap
    (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift f.toHom g.toHom
        (PullbackCone.mk s.fst.toHom s.snd.toHom sorry) ≫
      (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeft f.toHom g.toHom).snd)
    x)
  (CategoryStruct.comp (Y :=
    Y.presheaf.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (X :=
            (↑(Cone.pt (F := cospan f.toHom g.toHom) (PullbackCone.mk s.fst.toHom s.snd.toHom sorry)) : TopCat)) (Y :=
            (↑Y.toPresheafedSpace : TopCat))
            (PresheafedSpace.Hom.base (PullbackCone.snd (PullbackCone.mk s.fst.toHom s.snd.toHom sorry))) :
          (a : (↑(↑(PullbackCone.mk s.fst.toHom s.snd.toHom sorry).pt : TopCat) : Type u_1)) →
            (↑(↑Y.toPresheafedSpace : TopCat) : Type u_1))
        x))
    (eqToHom sorry) (PresheafedSpace.Hom.stalkMap (PullbackCone.snd (PullbackCone.mk s.fst.toHom s.snd.toHom sorry)) x))
Tactic: have :=
        PresheafedSpace.stalkMap.congr_hom _ _
          (PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_snd f.1 g.1
            (PullbackCone.mk s.fst.1 s.snd.1
              (congr_arg LocallyRingedSpace.Hom.toHom s.condition)))
          x