theorem extracted_formal_statement_52.{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}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω) (i : ι) :
  (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω k)) - ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ) =
    (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω k)) ∂Measure.pi fun x => μ) -
      ∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω
i : ι
⊢ (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω k)) - ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ) =
    (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω k)) ∂Measure.pi fun x => μ) -
      ∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ

Tactic: apply integral_sub
    · simp
    ·
  apply integrable_finset_sum
  exact fun i_1 a ↦ integrable_aux0 hf hf' i i_1

State after:

Relevant lemmas:
MeasureTheory.integrable_finset_sum : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] {ι : Type u_6} (s : Finset ι) {f : ι → α → β},
  (∀ i ∈ s, Integrable (f i) μ) → Integrable (fun a => ∑ i ∈ s, f i a) μ
MeasureTheory.integral_sub : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_2 : 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
integrable_aux0 : ∀ {Ω : Type u} [inst : 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},
  (∀ (i : ι), Measurable (f i ∘ X)) →
    ∀ {b : ℝ},
      (∀ (i : ι) (z : Z), |f i z| ≤ b) →
        ∀ (i : ι) (k : Fin n), Integrable (fun ω => f i (X (ω k))) (Measure.pi fun x => μ)