theorem extracted_formal_statement_205.{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} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω' : Fin n → Ω) (i : ι)
  (h_original_goal :
    |(|∫ (x : Fin n → Ω), ∑ k, (f i (X (ω' k)) - f i (X (x k))) ∂Measure.pi fun x => μ|)| ≤ ↑n * (b + b)) :
  |∫ (x : Fin n → Ω), ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)) ∂Measure.pi fun x => μ| ≤ ↑n * (b + b) := sorry

State before:
case hf.right.h.hf
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
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω' : Fin n → Ω
i : ι
⊢ |(|∫ (x : Fin n → Ω), ∑ k, (f i (X (ω' k)) - f i (X (x k))) ∂Measure.pi fun x => μ|)| ≤ ↑n * (b + b)

Tactic: simp only [Finset.sum_sub_distrib, abs_abs]

State after:
[case hf.right.h.hf
 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
 inst_2 : Nonempty ι
 inst_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω' : Fin n → Ω
 i : ι
 ⊢ |∫ (x : Fin n → Ω), ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)) ∂Measure.pi fun x => μ| ≤ ↑n * (b + b)]

Relevant lemmas:
abs_abs : ∀ {α : Type u_1} [inst : Lattice α] [inst_4 : AddGroup α] [inst_5 : AddLeftMono α] [inst_6 : AddRightMono α]
  (a : α), |(|a|)| = |a|
Finset.sum_sub_distrib : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} {f g : α → β} [inst : SubtractionCommMonoid β],
  ∑ x ∈ s, (f x - g x) = ∑ x ∈ s, f x - ∑ x ∈ s, g x