∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
  {𝕜' : Type u_2} [inst_5 : NormedField 𝕜'] [inst_6 : NormedAlgebra 𝕜 𝕜'] [inst_7 : NormedSpace 𝕜' F]
  [IsScalarTower 𝕜 𝕜' F] {f : E → 𝕜'} {g : E → F} {N : WithTop ℕ∞},
  ContDiffOn 𝕜 N f s →
    ContDiffOn 𝕜 N g s →
      UniqueDiffOn 𝕜 s →
        ∀ {x : E},
          x ∈ s →
            ∀ {n : ℕ},
              ↑n ≤ N →
                ‖iteratedFDerivWithin 𝕜 n (fun y => f y • g y) s x‖ ≤
                  ∑ i ∈ Finset.range (n + 1),
                    ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖