-- Mathlib/MeasureTheory/Function/Jacobian.lean
theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ ∑' (n : ℕ), (μ (f '' (s ∩ t n)) + ↑ε * μ (s ∩ t n) + ↑ε * μ (s ∩ t n)) =
    ∑' (a : ℕ), (μ (f '' (s ∩ t a)) + 2 * ↑ε * μ (s ∩ t a))

Tactic:
congr 1

Result:
case e_f
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ (fun n => μ (f '' (s ∩ t n)) + ↑ε * μ (s ∩ t n) + ↑ε * μ (s ∩ t n)) = fun a =>
    μ (f '' (s ∩ t a)) + 2 * ↑ε * μ (s ∩ t a)

Proof state:
case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
⊢ ∀ᵐ (a : E) ∂μ.restrict (s ∩ t n), ENNReal.ofReal |(f' a).det| ≤ ENNReal.ofReal |(A n).det| + ↑ε

Tactic:
filter_upwards [(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f' fun x hx =>
    (hf' x hx.1).mono inter_subset_left]

Result:
case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
⊢ ∀ (a : E), ‖f' a - A n‖₊ ≤ δ (A n) → ENNReal.ofReal |(f' a).det| ≤ ENNReal.ofReal |(A n).det| + ↑ε

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
this :
  ∀ (A : E →L[ℝ] E),
    ∃ δ,
      0 < δ ∧
        (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |B.det - A.det| ≤ ↑ε) ∧
          ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s) + 2 * ↑ε * μ s

Tactic:
choose δ hδ using this

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s) + 2 * ↑ε * μ s

Proof state:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
⊢ ∃ δ,
    0 < δ ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t

Tactic:
let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩

Result:
case intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
⊢ ∃ δ,
    0 < δ ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t