theorem extracted_formal_statement_138.{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)
  (h_original_goal : BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)) :
  ∃ x, ∀ y ∈ Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x := sorry

State before:
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
⊢ BddAbove (Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|)

Tactic: rw [bddAbove_def]

State after:
[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
 ⊢ ∃ x, ∀ y ∈ Set.range fun i => |∑ k, ↑↑(σ k) * (f i (X (ω k).1) - f i (X (ω k).2))|, y ≤ x]

Relevant lemmas:
bddAbove_def : ∀ {α : Type u} [inst : Preorder α] {s : Set α}, BddAbove s ↔ ∃ x, ∀ y ∈ s, y ≤ x