theorem extracted_formal_statement_142.{u_1, w, u} {Ω : Type u} {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  [inst : Countable ι] {X : Ω → Z} {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : Fin n → Ω × Ω) (σ : Signs n)
  (this : ∃ x, ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x) (z : ℝ)
  (hz : ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ z) (y : ℝ)
  (hy : ∃ y_1, |∑ k, ↑↑(σ k) * (f y_1 (X (ω k).1) - f y_1 (X (ω k).2))| = y) (i : ι)
  (hi : |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| = y) :
  ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ∈
    Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| := sorry

State before:
case a
Ω : Type u
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
inst✝ : Countable ι
X : Ω → Z
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω × Ω
σ : Signs n
this : ∃ x, ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x
z : ℝ
hz : ∀ y ∈ Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ z
y : ℝ
hy : ∃ y_1, |∑ k, ↑↑(σ k) * (f y_1 (X (ω k).1) - f y_1 (X (ω k).2))| = y
i : ι
hi : |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| = y
⊢ ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))| ∈
    Set.range fun i => ∑ k, |↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|

Tactic: simp

State after:

Relevant lemmas: