theorem extracted_formal_statement_217.{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' : Fin n → Ω) (i : ι) (this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)))
  (a : Fin n → Ω) :
  |∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (a x_1))| ≤ |∑ x, f i (X (ω' x))| + |∑ x_1, f i (X (a x_1))| := sorry

State before:
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' : Fin n → Ω
i : ι
this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1))
a : Fin n → Ω
⊢ |∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (a x_1))| ≤ |∑ x, f i (X (ω' x))| + |∑ x_1, f i (X (a x_1))|

Tactic: exact abs_sub (∑ x : Fin n, f i (X (ω' x))) (∑ x_1 : Fin n, f i (X (a x_1)))

State after:

Relevant lemmas:
abs_sub : ∀ {G : Type u_1} [inst : LinearOrderedAddCommGroup G] (a b : G), |a - b| ≤ |a| + |b|