theorem extracted_formal_statement_6.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} [inst_1 : MeasurableSpace 𝒳] [inst_2 : Nonempty 𝒳] [inst_3 : Nonempty ι] [inst_4 : Countable ι]
  [inst_5 : IsProbabilityMeasure μ] (f : ι → 𝒳 → ℝ) (hf : ∀ (i : ι), Measurable (f i)) (X : Ω → 𝒳) (hX : Measurable X)
  {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) {t : ℝ} (ht : 0 ≤ t) (ht' : t * b ^ 2 ≤ 1 / 2) {ε : ℝ}
  (hε : 0 ≤ ε) (hn_1 : ¬n = 0) (hn : 0 < n)
  (h_original_goal :
    ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
      ((Measure.pi fun x => μ) fun ω =>
          (uniformDeviation n f μ X (X ∘ ω) -
              ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≥
            ε).toReal) :
  ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ≤
    (Measure.pi fun x => μ) fun ω =>
      ε ≤
        uniformDeviation n f μ X (X ∘ ω) -
          ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ := sorry

State before:
n : ℕ
Ω : Type u
inst✝⁵ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
inst✝⁴ : MeasurableSpace 𝒳
inst✝³ : Nonempty 𝒳
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
f : ι → 𝒳 → ℝ
hf : ∀ (i : ι), Measurable (f i)
X : Ω → 𝒳
hX : Measurable X
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
t : ℝ
ht : 0 ≤ t
ht' : t * b ^ 2 ≤ 1 / 2
ε : ℝ
hε : 0 ≤ ε
hn✝ : ¬n = 0
hn : 0 < n
⊢ ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
    ((Measure.pi fun x => μ) fun ω =>
        (uniformDeviation n f μ X (X ∘ ω) -
            ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≥
          ε).toReal

Tactic: simp only [ge_iff_le, ne_eq, measure_ne_top, not_false_eq_true, ENNReal.toReal_le_toReal]

State after:
[n : ℕ
 Ω : Type u
 inst✝⁵ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 inst✝⁴ : MeasurableSpace 𝒳
 inst✝³ : Nonempty 𝒳
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 f : ι → 𝒳 → ℝ
 hf : ∀ (i : ι), Measurable (f i)
 X : Ω → 𝒳
 hX : Measurable X
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 t : ℝ
 ht : 0 ≤ t
 ht' : t * b ^ 2 ≤ 1 / 2
 ε : ℝ
 hε : 0 ≤ ε
 hn✝ : ¬n = 0
 hn : 0 < n
 ⊢ ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ≤
     (Measure.pi fun x => μ) fun ω =>
       ε ≤
         uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ]

Relevant lemmas:
ne_eq : ∀ {α : Sort u_1} (a b : α), (a ≠ b) = ¬a = b
not_false_eq_true : (¬False) = True
MeasureTheory.measure_ne_top : ∀ {α : Type u_1} {m0 : MeasurableSpace α} (μ : Measure α) [inst : IsFiniteMeasure μ]
  (s : Set α), μ s ≠ ⊤
ge_iff_le : ∀ {α : Type u} [inst : LE α] {x y : α}, x ≥ y ↔ y ≤ x
ENNReal.toReal_le_toReal : ∀ {a b : ℝ≥0∞}, a ≠ ⊤ → b ≠ ⊤ → (a.toReal ≤ b.toReal ↔ a ≤ b)