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 := by
  /- To bound `∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ`, we cover `s` by sets where `f` is
    well-approximated by linear maps `A n` (and where `f'` is almost everywhere close to `A n`),
    and then use that `f` expands the measure of such a set by at least `(A n).det - ε`. -/
  have :
    ∀ A : E →L[ℝ] E,
      ∃ δ : ℝ≥0,
        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 := by
    intro A
    obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε :=
      continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt ε εpos
    let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩
    have I'' : ∀ B : E →L[ℝ] E, ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε := by
      intro B hB
      rw [← Real.dist_eq]
      apply (hδ' B _).le
      rw [dist_eq_norm]
      exact hB.trans_lt (half_lt_self δ'pos)
    rcases eq_or_ne A.det 0 with (hA | hA)
    · refine' ⟨δ'', half_pos δ'pos, I'', _⟩
      simp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff,
        zero_le, abs_zero]
    let m : ℝ≥0 := Real.toNNReal |A.det| - ε
    have I : (m : ℝ≥0∞) < ENNReal.ofReal |A.det| := by
      simp only [ENNReal.ofReal, ENNReal.coe_sub]
      apply ENNReal.sub_lt_self ENNReal.coe_ne_top
      · simpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA
      · simp only [εpos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff]
    rcases ((mul_le_addHaar_image_of_lt_det μ A I).and self_mem_nhdsWithin).exists with ⟨δ, h, δpos⟩
    refine' ⟨min δ δ'', lt_min δpos (half_pos δ'pos), _, _⟩
    · intro B hB
      apply I'' _ (hB.trans _)
      simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]
    · intro t g htg
      rcases eq_or_ne (μ t) ∞ with (ht | ht)
      · simp only [ht, εpos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne.def,
          not_false_iff, _root_.add_top]
      have := h t g (htg.mono_num (min_le_left _ _))
      rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this
      simp only [ht, imp_true_iff, Ne.def, not_false_iff]
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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 → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧
          ∀ (t : Set E) (g : E → E),
            ApproximatesLinearOn g A t δ →
              ENNReal.ofReal |ContinuousLinearMap.det A| * ↑↑μ t ≤ ↑↑μ (g '' t) + ↑ε * ↑↑μ t
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ∂μ ≤ ↑↑μ (f '' s) + 2 * ↑ε * ↑↑μ s

Next tactic:
choose δ hδ using this