theorem extracted_formal_statement_74.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ω : Ω) (x : ℝ) (h : x ∈ Metric.ball t (|t| + 1))
  (h_original_goal : |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) :
  |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2 := sorry

State before:
case h_bound
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
ω : Ω
x : ℝ
h : x ∈ Metric.ball t (|t| + 1)
⊢ |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2

Tactic: simp only [sq_abs]

State after:
[case h_bound
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 c : ℝ := ‖a‖ ⊔ ‖b‖
 e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
 ω : Ω
 x : ℝ
 h : x ∈ Metric.ball t (|t| + 1)
 ⊢ |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2]

Relevant lemmas:
sq_abs : ∀ {α : Type u_1} [inst : LinearOrderedRing α] (a : α), |a| ^ 2 = a ^ 2