-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun (j : ℕ) ↦
    HSMul.hSMul (β := A⁰_ f) ((m + m).choose j)
      (if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
                    (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) j *
                      HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m - j))
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ })
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ })
        else
          HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ })
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
                    (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) (j - m) *
                      HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m + m - j))
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ }))
j : ℕ
hj : j ∈ range (m + m + 1)
⊢ Localization.mk
      ((m + m).choose j •
        (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) j *
          HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m + m - j)))
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i + i)) ⋯) =
    HomogeneousLocalization.val (g j)
have rw : Localization.mk
    ((m + m).choose j •
      (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) j *
        HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m + m - j)))
    (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ (i + i)) sorry) =
  HSMul.hSMul (β := Localization (Submonoid.powers f)) ((m + m).choose j)
    (HomogeneousLocalization.val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      (if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
                    (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) j *
                      HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m - j))
                    sorry,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry :
                NumDenSameDeg 𝒜 (Submonoid.powers f) })
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) m)
                    sorry,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry :
                NumDenSameDeg 𝒜 (Submonoid.powers f) })
        else
          HMul.hMul (α := HomogeneousLocalization 𝒜 (Submonoid.powers f))
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) m)
                    sorry,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry :
                NumDenSameDeg 𝒜 (Submonoid.powers f) })
            (HomogeneousLocalization.mk
              { deg := m * i,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
                    (HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) a : A) (j - m) *
                      HPow.hPow (α := A) ((proj 𝒜 i : (a : A) → A) b : A) (m + m - j))
                    sorry,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) sorry, den_mem := sorry :
                NumDenSameDeg 𝒜 (Submonoid.powers f) })))
Tactic: rw [HomogeneousLocalization.val_smul]