-- Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem
Goal:
E : Type u
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
n : ℕ
inst_2 : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I) x
Hd :
  ∀ (x : Fin (n + 1) → ℝ),
    Membership.mem (γ := Set (Fin (n + 1) → ℝ))
        ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I \ s : Set (Fin (n + 1) → ℝ)) x →
      HasFDerivWithinAt f (f' x) ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I) x
i : Fin (n + 1)
Hc : ContinuousOn f ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I)
fI : ℝ → Box (Fin n) → E :=
  fun (y : ℝ) (J : Box (Fin n)) ↦
    integral J GP (fun (x : Fin n → ℝ) ↦ f (i.insertNth (α := fun (a : Fin (n + 1)) ↦ ℝ) y x)) BoxAdditiveMap.volume
fb : (↑(Set.Icc (I.lower i) (I.upper i)) : Type) → Fin n →ᵇᵃ[(↑(I.face i) : WithTop (Box (Fin n)))] E :=
  fun (x : (↑(Set.Icc (I.lower i) (I.upper i)) : Type)) ↦
    Integrable.toBoxAdditive (vol := BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume)) ⋯
F : Fin (n + 1) →ᵇᵃ[(↑I : WithTop (Box (Fin (n + 1))))] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx :
  Membership.mem (γ := Set (Fin (n + 1) → ℝ))
    ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I ∩ s : Set (Fin (n + 1) → ℝ)) x
ε : ℝ
ε0 : ε > 0
this :
  ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    δ ∈ Set.Ioc 0 (1 / 2) ∧
      (∀ y₁ ∈ closedBall x δ ∩ (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I,
          ∀ y₂ ∈ closedBall x δ ∩ (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I,
            ‖f y₁ - f y₂‖ ≤ ε / 2) ∧
        (2 * δ) ^ (n + 1) *
            norm (E := E) ((f' x : (a : Fin (n + 1) → ℝ) → E) (Pi.single (M := fun (a : Fin (n + 1)) ↦ ℝ) i 1) : E) ≤
          ε / 2
refine_3 :
  ∀ (δ : ℝ) (hδ0 : 0 < δ) (hδ12 : δ ≤ 1 / 2)
    (hdfδ :
      ∀ y₁ ∈ closedBall x δ ∩ (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I,
        ∀ y₂ ∈ closedBall x δ ∩ (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) I, ‖f y₁ - f y₂‖ ≤ ε / 2)
    (hδ :
      (2 * δ) ^ (n + 1) *
          norm (E := E) ((f' x : (a : Fin (n + 1) → ℝ) → E) (Pi.single (M := fun (a : Fin (n + 1)) ↦ ℝ) i 1) : E) ≤
        ε / 2),
    ∃ δ > 0,
      ∀ J ≤ I,
        (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) J ⊆ closedBall x δ →
          Membership.mem (γ := Set (Fin (n + 1) → ℝ))
              ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) J : Set (Fin (n + 1) → ℝ)) x →
            (GP.bDistortion = true → J.distortion ≤ c) →
              dist (α := E)
                  ((((BoxAdditiveMap.volume : (a : Box (Fin (n + 1))) → E →L[ℝ] E) J : E →L[ℝ] E) : (a : E) → E)
                      ((f' x : (a : Fin (n + 1) → ℝ) → E) (Pi.single (M := fun (a : Fin (n + 1)) ↦ ℝ) i 1) : E) :
                    E)
                  ((F : (a : Box (Fin (n + 1))) → E) J : E) ≤
                ε
⊢ ∃ δ > 0,
    ∀ J ≤ I,
      (Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) J ⊆ closedBall x δ →
        Membership.mem (γ := Set (Fin (n + 1) → ℝ))
            ((Box.Icc : (a : Box (Fin (n + 1))) → Set (Fin (n + 1) → ℝ)) J : Set (Fin (n + 1) → ℝ)) x →
          (GP.bDistortion = true → J.distortion ≤ c) →
            dist (α := E)
                ((((BoxAdditiveMap.volume : (a : Box (Fin (n + 1))) → E →L[ℝ] E) J : E →L[ℝ] E) : (a : E) → E)
                    ((f' x : (a : Fin (n + 1) → ℝ) → E) (Pi.single (M := fun (a : Fin (n + 1)) ↦ ℝ) i 1) : E) :
                  E)
                ((F : (a : Box (Fin (n + 1))) → E) J : E) ≤
              ε

Tactic:
rcases this.exists with ⟨δ, ⟨hδ0, hδ12⟩, hdfδ, hδ⟩

Premises used: Filter.Eventually.exists