theorem extracted_formal_statement_19.{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] [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) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) (ω' : Ω × Ω) :
  |∫ (x : Fin n → Ω × Ω),
        2⁻¹ ^ n *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
    ↑n * (b + b) + (b + b + C) := sorry

State before:
case right.h
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
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
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
ω' : Ω × Ω
⊢ |∫ (x : Fin n → Ω × Ω),
        2⁻¹ ^ n *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ i) * (f I (X (x i).1) - f I (X (x i).2)) +
                (f I (X ω'.1) - f I (X ω'.2) + c I) ∂Measure.pi fun x => μ.prod μ| ≤
    ↑n * (b + b) + (b + b + C)

Tactic: apply abs_expectation_le_of_abs_le_const
    filter_upwards with ω
    exact bound_σsum h𝓕' (boundedness₀ h𝓕' hC ω') ω

State after:

Relevant lemmas:
abs_expectation_le_of_abs_le_const : ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_4 : IsProbabilityMeasure μ] {f : Ω → ℝ} {C : ℝ}, (∀ᵐ (x : Ω) ∂μ, |f x| ≤ C) → |∫ (x : Ω), f x ∂μ| ≤ C
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
bound_σsum : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} {X : Ω → Z} {n : ℕ} [inst : Nonempty ι] {b : ℝ},
  (∀ (I : ι) (z : Z), |f I z| ≤ b) →
    ∀ {c : ι → ℝ} {C : ℝ},
      (∀ (I : ι), |c I| ≤ C) →
        ∀ (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| ≤ ↑n * (b + b) + C