∀ {R : Type u_2} {A : Type u_1} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : IsDedekindDomain A] {I : Ideal R}
  {J : Ideal A} [inst_3 : IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J) (hI : I ≠ ⊥) (hJ : J ≠ ⊥) (j : Ideal A)
  (hj : j ∈ {M | M ∈ UniqueFactorizationMonoid.normalizedFactors J}),
  (fun j => ⟨↑((idealFactorsEquivOfQuotEquiv f) ⟨↑j, ⋯⟩), ⋯⟩)
      ((fun j => ⟨↑((idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, ⋯⟩), ⋯⟩) ⟨j, hj⟩) =
    ⟨j, hj⟩