theorem extracted_formal_statement_59.{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))|)| ≤ ↑n * b := sorry

State before:
case h.hf
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))|)| ≤ ↑n * b

Tactic: simp only [Int.reduceNeg, abs_abs]
    have l1 : |∑ k : Fin n, (i k : ℝ) * f i_1 (X (a.2 k))| ≤ ↑n * b := by
  calc
    _ ≤ ∑ k : Fin n, |(i k : ℝ) * f i_1 (X (a.2 k))| := by apply Finset.abs_sum_le_sum_abs
    _ ≤ ∑ k : Fin n, b := by
      apply Finset.sum_le_sum
      intro i_2 hi_2
      rw [abs_mul]
      rw [abs_sigma (i i_2)]
      simp only [one_mul, ge_iff_le]
      exact hf' i_1 (X (a.2 i_2))
    _ ≤ ↑n * b := by simp
    exact l1

State after:

Relevant lemmas:
error