theorem extracted_formal_statement_38.{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 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (y : Ω) (x : ℝ) (b1 : x ∈ Metric.ball t (|t| + 1)) : rexp |x * X y| * |X y| = rexp (|x| * |X y|) * |X y| := sorry

State before:
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)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω|
y : Ω
x : ℝ
b1 : x ∈ Metric.ball t (|t| + 1)
⊢ rexp |x * X y| * |X y| = rexp (|x| * |X y|) * |X y|

Tactic: rw [abs_mul x (X y)]

State after:

Relevant lemmas:
abs_mul : ∀ {α : Type u_1} [inst : LinearOrderedRing α] (a b : α), |a * b| = |a| * |b|