∀ {𝕜 : Type u_1} {ι : Type u_2} {F : Type u_4} {G : Type u_5} {H : Type u_6} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G]
  [inst_5 : NormedAddCommGroup H] [inst_6 : NormedSpace 𝕜 H] [inst_7 : Fintype ι] [inst_8 : DecidableEq ι]
  (fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)),
  HasStrictFDerivAt (fun fg => fg.1.compContinuousLinearMap fg.2)
    ((ContinuousAlternatingMap.compContinuousLinearMapCLM fg.2).comp
        (ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)) +
      (fg.1.fderivCompContinuousLinearMap fg.2).comp (ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)))
    fg