theorem extracted_formal_statement_232.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) :
  let f' := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2;
  ∀ (ω : Fin n → Ω × Ω),
    let V := fun f => ∑ i, (f (X (ω i).1) - f (X (ω i).2));
    (∀ (f : Z → ℝ), V (-f) = -V f) →
      BddAbove (Set.range fun i => |V (f i)|) →
        ⨆ I, |∑ i, (f I (X (ω i).1) - f I (X (ω i).2))| = ⨆ I, ∑ i, (f' I (X (ω i).1) - f' I (X (ω i).2)) := sorry

State before:
case e_f.h
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
f' : Fin 2 × ι → Z → ℝ := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2
ω : Fin n → Ω × Ω
V : (Z → ℝ) → ℝ := fun f => ∑ i, (f (X (ω i).1) - f (X (ω i).2))
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
⊢ ⨆ I, |∑ i, (f I (X (ω i).1) - f I (X (ω i).2))| = ⨆ I, ∑ i, (f' I (X (ω i).1) - f' I (X (ω i).2))

Tactic: exact sup_abs_lemma hV₀ hV₁

State after:

Relevant lemmas:
sup_abs_lemma : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι] {V : (Z → ℝ) → ℝ},
  (∀ (f : Z → ℝ), V (-f) = -V f) →
    BddAbove (Set.range fun i => |V (f i)|) → ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)