∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {D : Type uD} [inst_1 : NormedAddCommGroup D]
  [inst_2 : NormedSpace 𝕜 D] {E : Type uE} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {F : Type uF}
  [inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace 𝕜 F] {G : Type uG} [inst_7 : NormedAddCommGroup G]
  [inst_8 : NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : WithTop ℕ∞} {s : Set D} {x : D},
  ContDiffOn 𝕜 N f s →
    ContDiffOn 𝕜 N g s →
      UniqueDiffOn 𝕜 s →
        x ∈ s →
          ∀ {n : ℕ},
            ↑n ≤ N →
              ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
                ‖B‖ *
                  ∑ i ∈ Finset.range (n + 1),
                    ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖