theorem extracted_formal_statement_3.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : Nonempty ι] [inst_2 : Countable ι] [inst_3 : IsProbabilityMeasure μ]
  (hn : 0 < n) (X : Ω → 𝒳) (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b)
  (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) (ω : Fin n → Ω)
  (h_original_goal :
    ↑n * uniformDeviation n f μ X (X ∘ ω) = ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) :
  ⨆ i, ↑n * |(↑n)⁻¹ * ∑ k, f i ((X ∘ ω) k) - ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| =
    ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| := sorry

State before:
case h
n : ℕ
Ω : Type u
inst✝³ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
hn : 0 < n
X : Ω → 𝒳
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
ω : Fin n → Ω
⊢ ↑n * uniformDeviation n f μ X (X ∘ ω) = ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|

Tactic: rw [uniformDeviation, Real.mul_iSup_of_nonneg (by norm_num)]

State after:
[case h
 n : ℕ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 hn : 0 < n
 X : Ω → 𝒳
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 ω : Fin n → Ω
 ⊢ ⨆ i, ↑n * |(↑n)⁻¹ * ∑ k, f i ((X ∘ ω) k) - ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| =
     ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|]

Relevant lemmas:
Real.mul_iSup_of_nonneg : ∀ {ι : Sort u_1} {r : ℝ}, 0 ≤ r → ∀ (f : ι → ℝ), r * ⨆ i, f i = ⨆ i, r * f i