-- Module: Mathlib.Analysis.Calculus.ContDiff.Bounds
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
Du : Type u
inst_1 : NormedAddCommGroup Du
inst_2 : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
  ∀ {Eu Fu Gu : Type u} [inst_9 : NormedAddCommGroup Eu] [inst_10 : NormedSpace 𝕜 Eu] [inst_11 : NormedAddCommGroup Fu]
    [inst_12 : NormedSpace 𝕜 Fu] [inst_13 : NormedAddCommGroup Gu] [inst_14 : NormedSpace 𝕜 Gu]
    (B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du → Eu} {g : Du → Fu} (hf : ContDiffOn 𝕜 (↑n : ℕ∞ω) f s)
    (hg : ContDiffOn 𝕜 (↑n : ℕ∞ω) g s),
    norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ Du) Gu)
        (iteratedFDerivWithin 𝕜 n
          (fun (y : Du) ↦ (((B : (a : Eu) → Fu →L[𝕜] Gu) (f y) : Fu →L[𝕜] Gu) : (a : Fu) → Gu) (g y)) s x) ≤
      ‖B‖ *
        ∑ i ∈ Finset.range (n + 1),
          (↑(n.choose i) : ℝ) *
              norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
            norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i)) ↦ Du) Fu)
              (iteratedFDerivWithin 𝕜 (n - i) g s x)
Eu Fu Gu : Type u
inst_3 : NormedAddCommGroup Eu
inst_4 : NormedSpace 𝕜 Eu
inst_5 : NormedAddCommGroup Fu
inst_6 : NormedSpace 𝕜 Fu
inst_7 : NormedAddCommGroup Gu
inst_8 : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑(n + 1) : ℕ∞ω) f s
hg : ContDiffOn 𝕜 (↑(n + 1) : ℕ∞ω) g s
In : HAdd.hAdd (α := ℕ∞ω) (↑n : ℕ∞ω) 1 ≤ (↑n.succ : ℕ∞ω)
i : ℕ
hi : i ∈ Finset.range (n + 1)
rw :
  (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i)) ↦ Du) (Du →L[𝕜] Fu))
        (iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x) =
    (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i).succ) ↦ Du) Fu)
        (iteratedFDerivWithin 𝕜 (n - i).succ g s x)
rw₁ rw₂ :
  (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i)) ↦ Du) (Du →L[𝕜] Fu))
        (iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x) =
    (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i)) ↦ Du) (Du →L[𝕜] Fu))
        (iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x)
⊢ (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n - i)) ↦ Du) (Du →L[𝕜] Fu))
        (iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x) =
    (↑(n.choose i) : ℝ) *
        norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin i) ↦ Du) Eu) (iteratedFDerivWithin 𝕜 i f s x) *
      norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n + 1 - i)) ↦ Du) Fu)
        (iteratedFDerivWithin 𝕜 (n + 1 - i) g s x)

Tactic:
rw [Nat.succ_sub (Nat.lt_succ_iff.1 (Finset.mem_range.1 hi)),
            ← norm_iteratedFDerivWithin_fderivWithin hs hx]

Premises used: Finset.mem_range, Nat.lt_succ_iff, Nat.succ_sub, Iff.mp, norm_iteratedFDerivWithin_fderivWithin