∀ {ι : Type u_1} {A : Type u_2} {σ : Type u_3} [inst : CommRing A] [inst_1 : SetLike σ A]
  [inst_2 : AddSubgroupClass σ A] [inst_3 : AddCommMonoid ι] [inst_4 : DecidableEq ι] (𝒜 : ι → σ)
  [inst_5 : GradedRing 𝒜] {e : ι} {f g : A} (hg : g ∈ 𝒜 e) {x : A} (hx : x = f * g) {d : ι} (n : ℕ) (hf : f ∈ 𝒜 d)
  (a : A) (ha : a ∈ 𝒜 (n • d)),
  (HomogeneousLocalization.awayMap 𝒜 hg hx) (HomogeneousLocalization.Away.mk 𝒜 hf n a ha) =
    HomogeneousLocalization.Away.mk 𝒜 ⋯ n (a * g ^ n) ⋯