∀ {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} [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] {f : E → F [⋀^ι]→L[𝕜] G} {f' : E →L[𝕜] F [⋀^ι]→L[𝕜] G}
  {g : ι → E → F} {g' : ι → E →L[𝕜] F} {s : Set E} {x : E} [inst_7 : Fintype ι] [inst_8 : DecidableEq ι],
  HasFDerivWithinAt f f' s x →
    (∀ (i : ι), HasFDerivWithinAt (g i) (g' i) s x) →
      HasFDerivWithinAt (fun x => (f x) fun x_1 => g x_1 x)
        ((ContinuousAlternatingMap.apply 𝕜 F G fun x_1 => g x_1 x).comp f' +
          ∑ i, ((f x).toContinuousLinearMap (fun x_1 => g x_1 x) i).comp (g' i))
        s x