theorem extracted_formal_statement_38.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (i : Fin n) (h1 : AEStronglyMeasurable (fun x => 1) μ)
  (h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ) (h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ)
  (h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ)
  (this :
    (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
      (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)))
  (h_original_goal : AEStronglyMeasurable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ) :
  AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ := sorry

State before:
case left
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
i : Fin n
h1 : AEStronglyMeasurable (fun x => 1) μ
h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
this :
  (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
    (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
⊢ AEStronglyMeasurable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic: rw [this]

State after:
[case left
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 i : Fin n
 h1 : AEStronglyMeasurable (fun x => 1) μ
 h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
 h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
 h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
 h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
 this :
   (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
     (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
 ⊢ AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ]

Relevant lemmas: