-- Lean 4 proof step from Mathlib.Topology.Sheaves.CommRingCat (goal)
Goal:
X : TopCat
F : Sheaf CommRingCat X
U V W UW VW : Opens (↑X : Type u)
e : W ≤ U ⊔ V
x :
  (↑(CommRingCat.of
        ↥(RingHom.eqLocus
            (RingHom.comp
              (CommRingCat.Hom.hom
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
              (RingHom.fst
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) :
                  Type u)
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) :
                  Type u)))
            (RingHom.comp
              (CommRingCat.Hom.hom
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
              (RingHom.snd
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) :
                  Type u)
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) :
                  Type u))))) :
    Type u)
y : (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) : Type u)
h₁ : UW = U ⊓ W
h₂ : VW = V ⊓ W
⊢ (ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
              (homOfLE e).op) :
          (a : (↑(F.obj.obj (op (U ⊔ V))) : Type u)) → (↑(F.obj.obj (op W)) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (F.objSupIsoProdEqLocus U V).inv :
              (a :
                  (↑(CommRingCat.of
                        ↥(((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op)).comp
                                (RingHom.fst (↑(F.obj.obj (op U)) : Type u) (↑(F.obj.obj (op V)) : Type u))).eqLocus
                            ((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op)).comp
                              (RingHom.snd (↑(F.obj.obj (op U)) : Type u) (↑(F.obj.obj (op V)) : Type u))))) :
                    Type u)) →
                (↑(F.obj.obj (op (U ⊔ V))) : Type u))
            x :
          (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊔ V))) :
            Type u)) =
      y ↔
    Eq (α :=
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op UW)) :
          Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U))
                (Y :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op UW))
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑(F.obj.obj (op U)) : Type u)) → (↑(F.obj.obj (op UW)) : Type u))
            (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).1 :
          (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op UW)) :
            Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W))
                (Y :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op UW))
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op UW)) : Type u))
            y :
          (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op UW)) :
            Type u)) ∧
      Eq (α :=
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op VW)) :
          Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V))
                (Y :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op VW))
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑(F.obj.obj (op V)) : Type u)) → (↑(F.obj.obj (op VW)) : Type u))
            (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).2 :
          (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op VW)) :
            Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W))
                (Y :=
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op VW))
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op VW)) : Type u))
            y :
          (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op VW)) :
            Type u))
have subst : (ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := CommRingCat.carrier)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map (homOfLE e).op) :
        (a : (↑(F.obj.obj (op (U ⊔ V))) : Type u)) → (↑(F.obj.obj (op W)) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (F.objSupIsoProdEqLocus U V).inv :
            (a :
                (↑(CommRingCat.of
                      ↥(((CommRingCat.Hom.hom (F.obj.map (homOfLE sorry).op)).comp
                              (RingHom.fst (↑(F.obj.obj (op U)) : Type u) (↑(F.obj.obj (op V)) : Type u))).eqLocus
                          ((CommRingCat.Hom.hom (F.obj.map (homOfLE sorry).op)).comp
                            (RingHom.snd (↑(F.obj.obj (op U)) : Type u) (↑(F.obj.obj (op V)) : Type u))))) :
                  Type u)) →
              (↑(F.obj.obj (op (U ⊔ V))) : Type u))
          x :
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊔ V))) :
          Type u)) =
    y ↔
  Eq (α :=
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) (Y :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W)))
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                (Quiver.Hom.op (homOfLE sorry))) :
            (a : (↑(F.obj.obj (op U)) : Type u)) → (↑(F.obj.obj (op (U ⊓ W))) : Type u))
          (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).1 :
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
          Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) (Y :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W)))
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                (Quiver.Hom.op (homOfLE sorry))) :
            (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op (U ⊓ W))) : Type u))
          y :
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
          Type u)) ∧
    Eq (α :=
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) (Y :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W)))
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                (Quiver.Hom.op (homOfLE sorry))) :
            (a : (↑(F.obj.obj (op V)) : Type u)) → (↑(F.obj.obj (op (V ⊓ W))) : Type u))
          (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).2 :
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
          Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) (Y :=
              F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W)))
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                (Quiver.Hom.op (homOfLE sorry))) :
            (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op (V ⊓ W))) : Type u))
          y :
        (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
          Type u))
Tactic: subst h₁ h₂
Cleared hypotheses: UW, VW, h₁, h₂