induction n generalizing x with
| zero =>
  rw [orderedIteratedFPiDeriv_succ_eq_comp_left]
  rw [fpideriv_def, orderedIteratedFPiDeriv_zero_eq_comp]
  rw [Function.comp_apply, continuousMultilinearCurryLeftEquiv_apply]
  change fderiv 𝕜 (⇑(LinearIsometryEquiv.symm (continuousMultilinearIsEmptyEquiv 𝕜 (fun i => M (Fin.tail js i)) F)) ∘
    f ∘ (Function.update x (js 0))) _ _ _ = _
  rw [LinearIsometryEquiv.comp_fderiv]
  rw [ContinuousLinearMap.comp_apply]
  rfl
| succ n ih =>
  have A (y) : orderedIteratedFPiDeriv 𝕜 (Fin.tail js) f y =
      (continuousMultilinearCurryRightEquiv 𝕜 (fun i : Fin (n + 1) => M (js (Fin.succ i))) F ∘
        orderedIteratedFPiDeriv 𝕜 (Fin.init (Fin.tail js)) (fpideriv 𝕜 (js (Fin.last (n + 1))) f)) y
  . ext m
    rw [ih]
    rfl
  rw [orderedIteratedFPiDeriv_succ_eq_comp_left]
  rw [Function.comp_apply, continuousMultilinearCurryLeftEquiv_apply]
  rw [fpideriv_def]
  simp
  simp_rw [A]
  change fderiv 𝕜 (
    ⇑(continuousMultilinearCurryRightEquiv 𝕜 (fun i => M (js (Fin.succ i))) F) ∘
    (orderedIteratedFPiDeriv 𝕜 (Fin.init (Fin.tail js)) (fpideriv 𝕜 (js (Fin.last (n + 1))) f) ∘ Function.update x (js 0)))
    _ _ _ = _
  rw [LinearIsometryEquiv.comp_fderiv]
  rfl
ι : Type u_1
𝕜 : Type u_2
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
M : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (M i)
inst✝² : (i : ι) → NormedSpace 𝕜 (M i)
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : ((i : ι) → M i) → F
js : Fin (n + 1) → ι
x : (i : ι) → M i
m : (i : Fin (n + 1)) → M (js i)
⊢ (orderedIteratedFPiDeriv 𝕜 js f x) m =
    ((orderedIteratedFPiDeriv 𝕜 (Fin.init js) (fpideriv 𝕜 (js (Fin.last n)) f) x) (Fin.init m)) (m (Fin.last n))
no goals