theorem extracted_formal_statement_210.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b)
  (this :
    (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
      ∫ (x : Fin n → Ω × Ω),
        (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ) :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
this :
  (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ x_1, ⨆ I, ∑ i, ↑↑(x_1 i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ
⊢ (∫ (x : Fin n → Ω × Ω), ⨆ I, ∑ x_1, f I (X (x x_1).1) - ∑ x_1, f I (X (x x_1).2) ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (2 ^ n)⁻¹ * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) ∂Measure.pi fun x => μ.prod μ

Tactic: exact this

State after:

Relevant lemmas: