specialize @ih hf_pred
case succ.intro.intro
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
f : ((i : ι) → M i) → F
f' : ((i : ι) → M i) → FormalMultilinearSeries 𝕜 ((i : ι) → M i) F
n : ℕ
ih :
  HasFTaylorSeriesUpTo (↑n + 1) f f' →
    ∀ {js : Fin n → ι} {x : (i : ι) → M i},
      HasFDerivAt (orderedIteratedFPiDeriv 𝕜 js f)
        (ContinuousLinearMap.comp
          (ContinuousMultilinearMap.compContinuousLinearMapL fun i => ContinuousLinearMap.single (js i))
          (ContinuousMultilinearMap.curryLeft (f' x (n + 1))))
        x
js : Fin (Nat.succ n) → ι
x : (i : ι) → M i
hf_pred : HasFTaylorSeriesUpTo (↑(Nat.succ n)) f f'
right✝ : ContinuousOn (fun x => f' x (Nat.succ n + 1)) Set.univ
hf_diff :
  ∀ (x : (i : ι) → M i),
    HasFDerivAt (fun y => f' y (Nat.succ n)) (ContinuousMultilinearMap.curryLeft (f' x (Nat.succ (Nat.succ n)))) x
⊢ HasFDerivAt (orderedIteratedFPiDeriv 𝕜 js f)
    (ContinuousLinearMap.comp
      (ContinuousMultilinearMap.compContinuousLinearMapL fun i => ContinuousLinearMap.single (js i))
      (ContinuousMultilinearMap.curryLeft (f' x (Nat.succ n + 1))))
    x
case succ.intro.intro
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
f : ((i : ι) → M i) → F
f' : ((i : ι) → M i) → FormalMultilinearSeries 𝕜 ((i : ι) → M i) F
n : ℕ
js : Fin (Nat.succ n) → ι
x : (i : ι) → M i
hf_pred : HasFTaylorSeriesUpTo (↑(Nat.succ n)) f f'
right✝ : ContinuousOn (fun x => f' x (Nat.succ n + 1)) Set.univ
hf_diff :
  ∀ (x : (i : ι) → M i),
    HasFDerivAt (fun y => f' y (Nat.succ n)) (ContinuousMultilinearMap.curryLeft (f' x (Nat.succ (Nat.succ n)))) x
ih :
  ∀ {js : Fin n → ι} {x : (i : ι) → M i},
    HasFDerivAt (orderedIteratedFPiDeriv 𝕜 js f)
      (ContinuousLinearMap.comp
        (ContinuousMultilinearMap.compContinuousLinearMapL fun i => ContinuousLinearMap.single (js i))
        (ContinuousMultilinearMap.curryLeft (f' x (n + 1))))
      x
⊢ HasFDerivAt (orderedIteratedFPiDeriv 𝕜 js f)
    (ContinuousLinearMap.comp
      (ContinuousMultilinearMap.compContinuousLinearMapL fun i => ContinuousLinearMap.single (js i))
      (ContinuousMultilinearMap.curryLeft (f' x (Nat.succ n + 1))))
    x