-- File: FoML/FoML/Rademacher.lean
Theorem statement:
theorem extracted_formal_statement_316.{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 ω' : Fin n → Ω)
  (h_original_goal : (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) ω' ≤ (fun a => ⨆ i, ∑ k, (b + b)) ω')
  (i : ι) : |(|∑ k, (f i (X (ω k)) - f i (X (ω' k)))|)| ≤ (fun a => ⨆ i, ∑ k, (b + b)) ω' := sorry

Tactic state before:
case refine_2.h.h
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 ω' : Fin n → Ω
⊢ (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) ω' ≤ (fun a => ⨆ i, ∑ k, (b + b)) ω'

Tactic: apply abs_iSup_le

Tactic state after:
[case refine_2.h.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 ω' : Fin n → Ω
 ⊢ ∀ (i : ι), |(|∑ k, (f i (X (ω k)) - f i (X (ω' k)))|)| ≤ (fun a => ⨆ i, ∑ k, (b + b)) ω']

Premises used:
abs_iSup_le : ∀ {ι : Type v} [inst : Nonempty ι] {f : ι → ℝ} {a : ℝ}, (∀ (i : ι), |f i| ≤ a) → |⨆ i, f i| ≤ a