-- Lean 4 proof step from Mathlib.Analysis.Calculus.ContDiff.Bounds (hypothesis)
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)
⊢ norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin (n + 1)) ↦ Du) Gu)
      (iteratedFDerivWithin 𝕜 (n + 1)
        (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 + 1),
        (↑((n + 1).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)
have I1 : norm (E := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ Du) (Du →L[𝕜] Gu))
    (iteratedFDerivWithin 𝕜 n
      (fun (y : Du) ↦
        (((precompR Du B : (a : Eu) → (Du →L[𝕜] Fu) →L[𝕜] Du →L[𝕜] Gu) (f y) : (Du →L[𝕜] Fu) →L[𝕜] Du →L[𝕜] Gu) :
            (a : Du →L[𝕜] Fu) → Du →L[𝕜] Gu)
          (fderivWithin 𝕜 g s 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 + 1 - i)) ↦ Du) Fu)
          (iteratedFDerivWithin 𝕜 (n + 1 - i) g s x)
Tactic: have I1 :
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s x‖ ≤
          ‖B‖ * ∑ i ∈ Finset.range (n + 1), n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
            ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖ := by
      calc
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s x‖ ≤
            ‖B.precompR Du‖ * ∑ i ∈ Finset.range (n + 1),
              n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
                ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ :=
          IH _ (hf.of_le (Nat.cast_le.2 (Nat.le_succ n))) (hg.fderivWithin hs In)
        _ ≤ ‖B‖ * ∑ i ∈ Finset.range (n + 1), n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
              ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ := by
            gcongr; exact B.norm_precompR_le Du
        _ = _ := by
          congr 1
          apply Finset.sum_congr rfl fun i hi => ?_
          rw [Nat.succ_sub (Nat.lt_succ_iff.1 (Finset.mem_range.1 hi)),
            ← norm_iteratedFDerivWithin_fderivWithin hs hx]