/-- Auxiliary lemma for the divergence theorem. -/
theorem norm_volume_sub_integral_face_upper_sub_lower_smul_le {f : (Fin (n + 1) → ℝ) → E}
    {f' : (Fin (n + 1) → ℝ) →L[ℝ] E} (hfc : ContinuousOn f (Box.Icc I)) {x : Fin (n + 1) → ℝ}
    (hxI : x ∈ (Box.Icc I)) {a : E} {ε : ℝ} (h0 : 0 < ε)
    (hε : ∀ y ∈ (Box.Icc I), ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0}
    (hc : I.distortion ≤ c) :
    ‖(∏ j, (I.upper j - I.lower j)) • f' (Pi.single i 1) -
      (integral (I.face i) ⊥ (f ∘ i.insertNth (α := fun _ ↦ ℝ) (I.upper i)) BoxAdditiveMap.volume -
        integral (I.face i) ⊥ (f ∘ i.insertNth (α := fun _ ↦ ℝ) (I.lower i))
          BoxAdditiveMap.volume)‖ ≤
      2 * ε * c * ∏ j, (I.upper j - I.lower j) := by
  -- Porting note: Lean fails to find `α` in the next line
  set e : ℝ → (Fin n → ℝ) → (Fin (n + 1) → ℝ) := i.insertNth (α := fun _ ↦ ℝ)
  /- **Plan of the proof**. The difference of the integrals of the affine function
    `fun y ↦ a + f' (y - x)` over the faces `x i = I.upper i` and `x i = I.lower i` is equal to the
    volume of `I` multiplied by `f' (Pi.single i 1)`, so it suffices to show that the integral of
    `f y - a - f' (y - x)` over each of these faces is less than or equal to `ε * c * vol I`. We
    integrate a function of the norm `≤ ε * diam I.Icc` over a box of volume
    `∏ j ≠ i, (I.upper j - I.lower j)`. Since `diam I.Icc ≤ c * (I.upper i - I.lower i)`, we get the
    required estimate.  -/
  have Hl : I.lower i ∈ Icc (I.lower i) (I.upper i) := Set.left_mem_Icc.2 (I.lower_le_upper i)
  have Hu : I.upper i ∈ Icc (I.lower i) (I.upper i) := Set.right_mem_Icc.2 (I.lower_le_upper i)
  have Hi : ∀ x ∈ Icc (I.lower i) (I.upper i),
      Integrable.{0, u, u} (I.face i) ⊥ (f ∘ e x) BoxAdditiveMap.volume := fun x hx =>
    integrable_of_continuousOn _ (Box.continuousOn_face_Icc hfc hx) volume
  /- We start with an estimate: the difference of the values of `f` at the corresponding points
    of the faces `x i = I.lower i` and `x i = I.upper i` is `(2 * ε * diam I.Icc)`-close to the
    value of `f'` on `Pi.single i (I.upper i - I.lower i) = lᵢ • eᵢ`, where
    `lᵢ = I.upper i - I.lower i` is the length of `i`-th edge of `I` and `eᵢ = Pi.single i 1` is the
    `i`-th unit vector. -/
  have : ∀ y ∈ Box.Icc (I.face i),
      ‖f' (Pi.single i (I.upper i - I.lower i)) -
          (f (e (I.upper i) y) - f (e (I.lower i) y))‖ ≤
        2 * ε * diam (Box.Icc I) := fun y hy ↦ by
    set g := fun y => f y - a - f' (y - x) with hg
    change ∀ y ∈ (Box.Icc I), ‖g y‖ ≤ ε * ‖y - x‖ at hε
    clear_value g; obtain rfl : f = fun y => a + f' (y - x) + g y := by simp [hg]
    convert_to ‖g (e (I.lower i) y) - g (e (I.upper i) y)‖ ≤ _
    · congr 1
      have := Fin.insertNth_sub_same (α := fun _ ↦ ℝ) i (I.upper i) (I.lower i) y
      simp only [← this, f'.map_sub]; abel
    · have : ∀ z ∈ Icc (I.lower i) (I.upper i), e z y ∈ (Box.Icc I) := fun z hz =>
        I.mapsTo_insertNth_face_Icc hz hy
      replace hε : ∀ y ∈ (Box.Icc I), ‖g y‖ ≤ ε * diam (Box.Icc I)
      · intro y hy
        refine' (hε y hy).trans (mul_le_mul_of_nonneg_left _ h0.le)
        rw [← dist_eq_norm]
        exact dist_le_diam_of_mem I.isCompact_Icc.isBounded hy hxI
      rw [two_mul, add_mul]
      exact norm_sub_le_of_le (hε _ (this _ Hl)) (hε _ (this _ Hu))
  calc
    ‖(∏ j, (I.upper j - I.lower j)) • f' (Pi.single i 1) -
            (integral (I.face i) ⊥ (f ∘ e (I.upper i)) BoxAdditiveMap.volume -
              integral (I.face i) ⊥ (f ∘ e (I.lower i)) BoxAdditiveMap.volume)‖ =
        ‖integral.{0, u, u} (I.face i) ⊥
            (fun x : Fin n → ℝ =>
              f' (Pi.single i (I.upper i - I.lower i)) -
                (f (e (I.upper i) x) - f (e (I.lower i) x)))
            BoxAdditiveMap.volume‖ := by
      rw [← integral_sub (Hi _ Hu) (Hi _ Hl), ← Box.volume_face_mul i, mul_smul, ← Box.volume_apply,
        ← BoxAdditiveMap.toSMul_apply, ← integral_const, ← BoxAdditiveMap.volume,
        ← integral_sub (integrable_const _) ((Hi _ Hu).sub (Hi _ Hl))]
      simp only [(· ∘ ·), Pi.sub_def, ← f'.map_smul, ← Pi.single_smul', smul_eq_mul, mul_one]
    _ ≤ (volume (I.face i : Set (Fin n → ℝ))).toReal * (2 * ε * c * (I.upper i - I.lower i)) := by
      -- The hard part of the estimate was done above, here we just replace `diam I.Icc`
      -- with `c * (I.upper i - I.lower i)`
      

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
hε : ∀ y ∈ Box.Icc I, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖
c : ℝ≥0
hc : Box.distortion I ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := Fin.insertNth i
Hl : Box.lower I i ∈ Set.Icc (Box.lower I i) (Box.upper I i)
Hu : Box.upper I i ∈ Set.Icc (Box.lower I i) (Box.upper I i)
Hi : ∀ x ∈ Set.Icc (Box.lower I i) (Box.upper I i), Integrable (Box.face I i) ⊥ (f ∘ e x) BoxAdditiveMap.volume
this :
  ∀ y ∈ Box.Icc (Box.face I i),
    ‖f' (Pi.single i (Box.upper I i - Box.lower I i)) - (f (e (Box.upper I i) y) - f (e (Box.lower I i) y))‖ ≤
      2 * ε * diam (Box.Icc I)
⊢ ‖integral (Box.face I i) ⊥
        (fun x =>
          f' (Pi.single i (Box.upper I i - Box.lower I i)) - (f (e (Box.upper I i) x) - f (e (Box.lower I i) x)))
        BoxAdditiveMap.volume‖ ≤
    ENNReal.toReal (↑↑volume ↑(Box.face I i)) * (2 * ε * ↑c * (Box.upper I i - Box.lower I i))

Next tactic:
refine' norm_integral_le_of_le_const (fun y hy => (this y hy).trans _) volume