theorem extracted_formal_statement_96.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t : ℝ) (f : ℝ → ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (ω : Ω)
  (w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ))
  (h_original_goal :
    ((∂μ.tilted fun ω => t * X ω/∂μ) ω).toReal • f (X ω) = (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)) :
  (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
    (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω) := sorry

State before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t : ℝ
f : ℝ → ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
ω : Ω
w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
⊢ ((∂μ.tilted fun ω => t * X ω/∂μ) ω).toReal • f (X ω) = (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)

Tactic: rw [w]

State after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 t : ℝ
 f : ℝ → ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 ω : Ω
 w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
 ⊢ (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
     (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)]

Relevant lemmas: