∀ {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {H : Type u_6}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
  [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G]
  [inst_7 : NormedAddCommGroup H] [inst_8 : NormedSpace 𝕜 H] {f : E → G [⋀^ι]→L[𝕜] H} {g : E → F →L[𝕜] G} {s : Set E}
  {x : E} [inst_9 : Fintype ι] [inst_10 : DecidableEq ι],
  DifferentiableWithinAt 𝕜 f s x →
    DifferentiableWithinAt 𝕜 g s x →
      UniqueDiffWithinAt 𝕜 s x →
        fderivWithin 𝕜 (fun x => (f x).compContinuousLinearMap (g x)) s x =
          (ContinuousAlternatingMap.compContinuousLinearMapCLM (g x)).comp (fderivWithin 𝕜 f s x) +
            ((f x).fderivCompContinuousLinearMap (g x)).comp (fderivWithin 𝕜 g s x)