theorem extracted_formal_statement_123.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] [inst_1 : NeZero μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ)
  (h_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) (t : ℝ)
  (r0 :
    (fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = fun t =>
      (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (r01 :
    (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (r0' :
    (∫ (x : Ω), X x ^ 2 ∂μ.tilted fun ω => t * X ω) =
      (∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (p :
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ =
      ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
        ((∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ))
  (p'' :
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
          (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
      ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
          (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) /
        mgf X μ t ^ 2) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (x : Ω), X x ∈ Set.Icc a b → |X x| ≤ |c| := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
g' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω
g'' : ℝ → ℝ :=
  fun t => (∫ (x : Ω), (X ^ 2) x ∂μ.tilted fun ω => t * X ω) - (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) ^ 2
t : ℝ
r0 :
  (fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = fun t =>
    (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
r01 : (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
r0' :
  (∫ (x : Ω), X x ^ 2 ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
p :
  (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ =
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
      ((∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ)
p'' :
  ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
        (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
      (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) /
      mgf X μ t ^ 2
c : ℝ := ‖a‖ ⊔ ‖b‖
x : Ω
h : X x ∈ Set.Icc a b
⊢ |X x| ≤ |c|

Tactic: exact le_trans' (le_abs_self (max ‖a‖ ‖b‖)) (abs_le_max_abs_abs h.1 h.2)

State after:

Relevant lemmas:
abs_le_max_abs_abs : ∀ {G : Type u_1} [inst : LinearOrderedAddCommGroup G] {a b c : G}, a ≤ b → b ≤ c → |b| ≤ |a| ⊔ |c|
le_trans' : ∀ {α : Type u_2} [inst : Preorder α] {a b c : α}, b ≤ c → a ≤ b → a ≤ c
le_abs_self : ∀ {α : Type u_1} [inst : Lattice α] [inst_2 : AddGroup α] (a : α), a ≤ |a|