theorem extracted_formal_statement_370.{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)
  (a : (Fin n → Ω) × (Fin n → Ω)) (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) :
  |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))| ≤ ∑ k, |↑↑(i k) * f i_1 (X (a.2 k))| := 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
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
a : (Fin n → Ω) × (Fin n → Ω)
i : Signs n
hi : i ∈ Finset.univ
i_1 : ι
⊢ |∑ k, ↑↑(i k) * f i_1 (X (a.2 k))| ≤ ∑ k, |↑↑(i k) * f i_1 (X (a.2 k))|

Tactic: apply Finset.abs_sum_le_sum_abs

State after:

Relevant lemmas:
Finset.abs_sum_le_sum_abs : ∀ {ι : Type u_1} {G : Type u_9} [inst : LinearOrderedAddCommGroup G] (f : ι → G)
  (s : Finset ι), |∑ i ∈ s, f i| ≤ ∑ i ∈ s, |f i|