/-- The functions satisfying `isLocallyFraction` form a subring.
-/
def sectionsSubring (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
    Subring (∀ x : U.unop, Localizations R x) where
  carrier := { f | (isLocallyFraction R).pred f }
  zero_mem' := by
    refine' fun x => ⟨unop U, x.2, 𝟙 _, 0, 1, fun y => ⟨_, _⟩⟩
    · rw [← Ideal.ne_top_iff_one]; exact y.1.IsPrime.1
    · simp
  one_mem' := by
    refine' fun x => ⟨unop U, x.2, 𝟙 _, 1, 1, fun y => ⟨_, _⟩⟩
    · rw [← Ideal.ne_top_iff_one]; exact y.1.IsPrime.1
    · simp
  add_mem' := by
    intro a b ha hb x
    rcases ha x with ⟨Va, ma, ia, ra, sa, wa⟩
    rcases hb x with ⟨Vb, mb, ib, rb, sb, wb⟩
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
U : (Opens ↑(PrimeSpectrum.Top R))ᵒᵖ
a b : (x : ↥U.unop) → Localizations R ↑x
ha :
  a ∈
    {
          toSubsemigroup :=
            { carrier := {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f},
              mul_mem' :=
                (_ :
                  ∀ {a b : (x : ↥U.unop) → Localizations R ↑x},
                    a ∈ {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f} →
                      b ∈ {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f} →
                        ∀ (x : ↥U.unop),
                          ∃ V,
                            ∃ (_ : ↑x ∈ V),
                              ∃ i,
                                PrelocalPredicate.pred (isFractionPrelocal R) fun x =>
                                  (a * b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) },
          one_mem' :=
            (_ :
              ∀ (x : ↥U.unop),
                ∃ V,
                  ∃ (_ : ↑x ∈ V),
                    ∃ i,
                      PrelocalPredicate.pred (isFractionPrelocal R) fun x =>
                        OfNat.ofNat 1
                          ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) }.toSubsemigroup.carrier
hb :
  b ∈
    {
          toSubsemigroup :=
            { carrier := {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f},
              mul_mem' :=
                (_ :
                  ∀ {a b : (x : ↥U.unop) → Localizations R ↑x},
                    a ∈ {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f} →
                      b ∈ {f | PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate f} →
                        ∀ (x : ↥U.unop),
                          ∃ V,
                            ∃ (_ : ↑x ∈ V),
                              ∃ i,
                                PrelocalPredicate.pred (isFractionPrelocal R) fun x =>
                                  (a * b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) },
          one_mem' :=
            (_ :
              ∀ (x : ↥U.unop),
                ∃ V,
                  ∃ (_ : ↑x ∈ V),
                    ∃ i,
                      PrelocalPredicate.pred (isFractionPrelocal R) fun x =>
                        OfNat.ofNat 1
                          ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) }.toSubsemigroup.carrier
x : ↥U.unop
Va : Opens ↑(PrimeSpectrum.Top R)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ra sa : R
wa :
  ∀ (x : ↥Va),
    sa ∉ (↑x).asIdeal ∧
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x *
          (algebraMap R (Localizations R ↑x)) sa =
        (algebraMap R (Localizations R ↑x)) ra
Vb : Opens ↑(PrimeSpectrum.Top R)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
rb sb : R
wb :
  ∀ (x : ↥Vb),
    sb ∉ (↑x).asIdeal ∧
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x *
          (algebraMap R (Localizations R ↑x)) sb =
        (algebraMap R (Localizations R ↑x)) rb
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal R) fun x =>
          (a + b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
refine' ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ra * sb + rb * sa, sa * sb, _⟩