theorem extracted_formal_statement_14.{u} {Ω : Type u} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} (s : Fin (n + 1) → Set Ω) (hs : ∀ (i : Fin (n + 1)), MeasurableSet (s i))
  (h_original_goal :
    (Measure.map (fun f x => Fin.snoc f.2 f.1 x) (μ.prod (Measure.pi fun x => μ))) (Set.univ.pi s) = ∏ i, μ (s i)) :
  (μ.prod (Measure.pi fun x => μ)) ((fun f => Fin.snoc f.2 f.1) ⁻¹' Set.univ.pi s) = ∏ i, μ (s i) := sorry

State before:
case h
Ω : Type u
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
s : Fin (n + 1) → Set Ω
hs : ∀ (i : Fin (n + 1)), MeasurableSet (s i)
⊢ (Measure.map (fun f x => Fin.snoc f.2 f.1 x) (μ.prod (Measure.pi fun x => μ))) (Set.univ.pi s) = ∏ i, μ (s i)

Tactic: rw [Measure.map_apply measurable_snocEquiv (MeasurableSet.univ_pi hs)]

State after:
[case h
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 n : ℕ
 s : Fin (n + 1) → Set Ω
 hs : ∀ (i : Fin (n + 1)), MeasurableSet (s i)
 ⊢ (μ.prod (Measure.pi fun x => μ)) ((fun f => Fin.snoc f.2 f.1) ⁻¹' Set.univ.pi s) = ∏ i, μ (s i)]

Relevant lemmas:
MeasurableSet.univ_pi : ∀ {δ : Type u_4} {X : δ → Type u_6} [inst : (a : δ) → MeasurableSpace (X a)]
  [inst_2 : Countable δ] {t : (i : δ) → Set (X i)}, (∀ (i : δ), MeasurableSet (t i)) → MeasurableSet (Set.univ.pi t)
MeasureTheory.Measure.map_apply : ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {μ : Measure α} {f : α → β}, Measurable f → ∀ {s : Set β}, MeasurableSet s → (Measure.map f μ) s = μ (f ⁻¹' s)
measurable_snocEquiv : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {n : ℕ}, Measurable fun f => Fin.snoc f.2 f.1