∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {A : Type u_3} [inst_3 : NormedRing A] [inst_4 : NormedAlgebra 𝕜 A] {f g : E → A}
  {N : WithTop ℕ∞},
  ContDiff 𝕜 N f →
    ContDiff 𝕜 N g →
      ∀ (x : E) {n : ℕ},
        ↑n ≤ N →
          ‖iteratedFDeriv 𝕜 n (fun y => f y * g y) x‖ ≤
            ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖