theorem extracted_formal_statement_242.{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) :
  let f' := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2;
  (∀ (I : Fin 2 × ι), Measurable (f' I ∘ X)) →
    (∀ (I : Fin 2 × ι) (z : Z), |f' I z| ≤ b) →
      (∫ (x : Fin n → Ω × Ω),
          (fun ω => ⨆ I, ∑ i, (f' I (X (ω i).1) - f' I (X (ω i).2))) x ∂Measure.pi fun x => μ.prod μ) =
        ∫ (x : Fin n → Ω × Ω),
          (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f' I (X (ω i).1) - f' I (X (ω i).2)))
            x ∂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
f' : Fin 2 × ι → Z → ℝ := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2
h𝓕₂ : ∀ (I : Fin 2 × ι), Measurable (f' I ∘ X)
h𝓕'₂ : ∀ (I : Fin 2 × ι) (z : Z), |f' I z| ≤ b
⊢ (∫ (x : Fin n → Ω × Ω), (fun ω => ⨆ I, ∑ i, (f' I (X (ω i).1) - f' I (X (ω i).2))) x ∂Measure.pi fun x => μ.prod μ) =
    ∫ (x : Fin n → Ω × Ω),
      (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f' I (X (ω i).1) - f' I (X (ω i).2)))
        x ∂Measure.pi fun x => μ.prod μ

Tactic: exact symmetrization_equation h𝓕₂ h𝓕'₂

State after:

Relevant lemmas:
symmetrization_equation : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} [inst : MeasurableSpace Ω]
  {X : Ω → Z} {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] {n : ℕ} [inst_5 : Countable ι] [inst_6 : Nonempty ι],
  (∀ (I : ι), Measurable (f I ∘ X)) →
    ∀ {b : ℝ},
      (∀ (I : ι) (z : Z), |f I z| ≤ b) →
        (∫ (x : Fin n → Ω × Ω),
            (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2))) x ∂Measure.pi fun x => μ.prod μ) =
          ∫ (x : Fin n → Ω × Ω),
            (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)))
              x ∂Measure.pi fun x => μ.prod μ