theorem extracted_formal_statement_329.{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) × (Fin n → Ω)) (i : ι) : |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))| ≤ ↑n * (b + b) := sorry

State before:
case h.hf
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) × (Fin n → Ω)
i : ι
⊢ |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))| ≤ ↑n * (b + b)

Tactic: exact bounded_difference_of_bounded hf' ω.1 i ω.2

State after:

Relevant lemmas:
bounded_difference_of_bounded : ∀ {Ω : Type u} {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ} {X : Ω → Z} {b : ℝ},
  (∀ (i : ι) (z : Z), |f i z| ≤ b) →
    ∀ (ω : Fin n → Ω) (i : ι) (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ≤ ↑n * (b + b)