theorem extracted_formal_statement_83.{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} (hb : b ≥ 0)
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (hn : ¬n = 0)
  (hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C)
  (hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y)
  (hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k))))
  (hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))) :
  ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
      (fun ω =>
          ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) +
            (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
        x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) =
    ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) +
      ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) := sorry

State before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : b ≥ 0
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
hn : ¬n = 0
hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
⊢ ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
      (fun ω =>
          ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) +
            (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
        x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) =
    ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) +
      ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ)

Tactic: apply integral_add
    · apply integrable_aux2 hf hb hf'
    ·
  rw [← integrable_swap_iff]
  apply integrable_aux2 hf hb hf'

State after:

Relevant lemmas:
MeasureTheory.integrable_swap_iff : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α]
  [inst_4 : MeasurableSpace β] {μ : Measure α} {ν : Measure β} [inst_5 : NormedAddCommGroup E] [inst_6 : SFinite ν]
  [inst_7 : SFinite μ] {f : α × β → E}, Integrable (f ∘ Prod.swap) (ν.prod μ) ↔ Integrable f (μ.prod ν)
integrable_aux2 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] [inst_5 : Countable ι] {X : Ω → Z},
  (∀ (i : ι), Measurable (f i ∘ X)) →
    ∀ {b : ℝ},
      0 ≤ b →
        (∀ (i : ι) (z : Z), |f i z| ≤ b) →
          Integrable (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)
            ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))
MeasureTheory.integral_add : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} {f g : α → G},
  Integrable f μ → Integrable g μ → ∫ (a : α), f a + g a ∂μ = ∫ (a : α), f a ∂μ + ∫ (a : α), g a ∂μ