theorem extracted_formal_statement_3.{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} {n : ℕ} (ω : Ω × Ω) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (ih :
    ∀ (c : ι → ℝ),
      (∃ C, ∀ (I : ι), |c I| ≤ C) →
        (∫ (ω' : Fin n → Ω × Ω), ⨆ I, ∑ i, (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I ∂Measure.pi fun x => μ.prod μ) =
          ∫ (ω' : Fin n → Ω × Ω),
            2⁻¹ ^ n *
              ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I ∂Measure.pi fun x => μ.prod μ) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  ∫ (x : Fin n → Ω × Ω),
      (fun ω' => ⨆ I, ∑ i, (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I)) x ∂μ2n =
    ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
        x ∂μ2n := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
n : ℕ
ω : Ω × Ω
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
ih :
  ∀ (c : ι → ℝ),
    (∃ C, ∀ (I : ι), |c I| ≤ C) →
      (∫ (ω' : Fin n → Ω × Ω), ⨆ I, ∑ i, (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I ∂Measure.pi fun x => μ.prod μ) =
        ∫ (ω' : Fin n → Ω × Ω),
          2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + c I ∂Measure.pi fun x => μ.prod μ
⊢ let μ2n := Measure.pi fun x => μ.prod μ;
  ∫ (x : Fin n → Ω × Ω),
      (fun ω' => ⨆ I, ∑ i, (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I)) x ∂μ2n =
    ∫ (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)) + (f I (X ω.1) - f I (X ω.2) + c I))
        x ∂μ2n

Tactic: apply ih (fun I ↦ (f I (X ω.1) - f I (X ω.2)) + c I)
    use b + b + C
    exact boundedness₀ h𝓕' hC ω

State after:

Relevant lemmas:
boundedness₀ : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} {X : Ω → Z} {b : ℝ},
  (∀ (I : ι) (z : Z), |f I z| ≤ b) →
    ∀ {c : ι → ℝ} {C : ℝ}, (∀ (I : ι), |c I| ≤ C) → ∀ (ω : Ω × Ω) (I : ι), |f I (X ω.1) - f I (X ω.2) + c I| ≤ b + b + C