theorem extracted_formal_statement_243.{w, v, u} {Z : Type w} {ι : Type v} {f_1 : 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_1 I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f_1 I z| ≤ b) (ω : Fin n → Ω × Ω) (σ : Signs n) (f : Z → ℝ)
  (h_original_goal :
    ∑ i, ↑↑(σ i) * (-f (X (ω i).1) - -f (X (ω i).2)) = -∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))) :
  ∑ i, ↑↑(σ i) * (-f (X (ω i).1) - -f (X (ω i).2)) = ∑ x, -(↑↑(σ x) * (f (X (ω x).1) - f (X (ω x).2))) := sorry

State before:
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
σ : Signs n
V : (Z → ℝ) → ℝ := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))
f : Z → ℝ
⊢ ∑ i, ↑↑(σ i) * (-f (X (ω i).1) - -f (X (ω i).2)) = -∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))

Tactic: rw [← Finset.sum_neg_distrib]

State after:
[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
 σ : Signs n
 V : (Z → ℝ) → ℝ := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))
 f : Z → ℝ
 ⊢ ∑ i, ↑↑(σ i) * (-f (X (ω i).1) - -f (X (ω i).2)) = ∑ x, -(↑↑(σ x) * (f (X (ω x).1) - f (X (ω x).2)))]

Relevant lemmas:
Finset.sum_neg_distrib : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} {f : α → β} [inst : SubtractionCommMonoid β],
  ∑ x ∈ s, -f x = -∑ x ∈ s, f x