∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {φ : S →ₐ[R] S}
  {Q : Ideal S} (H : φ.IsArithFrobAt Q) (x : S), H.restrict ((Ideal.Quotient.mk Q) x) = (Ideal.Quotient.mk Q) (φ x)