theorem extracted_formal_statement_310.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : Fin n → Ω)
  (this :
    (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
      ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ)
  (h : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)) :
  ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
    ∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω
this :
  (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
    ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ
h : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)
⊢ ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
    ∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ

Tactic: apply norm_integral_le_integral_norm

State after:

Relevant lemmas:
MeasureTheory.norm_integral_le_integral_norm : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G]
  [inst_4 : NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} (f : α → G),
  ‖∫ (a : α), f a ∂μ‖ ≤ ∫ (a : α), ‖f a‖ ∂μ