theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    simp only []
    -- Here, both sides of the equation are equal to a restriction of `s`
    trans
    convert congr_arg ((structureSheaf R).1.map iDj.op) (hs j).symm using 1
    convert congr_arg ((structureSheaf R).1.map iDi.op) (hs i) using 1; swap
    

Proof state:
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ ((structureSheaf R).val.map iDj.op)
      (const R (a j) (h j) (PrimeSpectrum.basicOpen (h j))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h j), y ∈ PrimeSpectrum.basicOpen (h j))) =
    const R (h i * a j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal)
case h.e'_2
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ const R (a i * h j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal) =
    ((structureSheaf R).val.map iDi.op)
      (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)))

Next tactic:
all_goals rw [res_const]; apply const_ext; ring