theorem extracted_formal_statement_21.{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))
  (this : (fun f x => Fin.snoc f.2 f.1 x) ⁻¹' Set.univ.pi s = s (Fin.last n) ×ˢ Set.univ.pi (s ∘ Fin.castSucc))
  (h_original_goal : (μ.prod (Measure.pi fun x => μ)) ((fun f => Fin.snoc f.2 f.1) ⁻¹' Set.univ.pi s) = ∏ i, μ (s i)) :
  μ (s (Fin.last n)) * ∏ i, μ ((s ∘ Fin.castSucc) i) = ∏ 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)
this : (fun f x => Fin.snoc f.2 f.1 x) ⁻¹' Set.univ.pi s = s (Fin.last n) ×ˢ Set.univ.pi (s ∘ Fin.castSucc)
⊢ (μ.prod (Measure.pi fun x => μ)) ((fun f => Fin.snoc f.2 f.1) ⁻¹' Set.univ.pi s) = ∏ i, μ (s i)

Tactic: rw [this, Measure.prod_prod, Measure.pi_pi]

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)
 this : (fun f x => Fin.snoc f.2 f.1 x) ⁻¹' Set.univ.pi s = s (Fin.last n) ×ˢ Set.univ.pi (s ∘ Fin.castSucc)
 ⊢ μ (s (Fin.last n)) * ∏ i, μ ((s ∘ Fin.castSucc) i) = ∏ i, μ (s i)]

Relevant lemmas:
MeasureTheory.Measure.pi_pi : ∀ {ι : Type u_1} {α : ι → Type u_3} [inst : Fintype ι]
  [inst_2 : (i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → Measure (α i)) [inst_3 : ∀ (i : ι), SigmaFinite (μ i)]
  (s : (i : ι) → Set (α i)), (Measure.pi μ) (Set.univ.pi s) = ∏ i, (μ i) (s i)
MeasureTheory.Measure.prod_prod : ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α]
  [inst_2 : MeasurableSpace β] {μ : Measure α} {ν : Measure β} [inst_3 : SFinite ν] (s : Set α) (t : Set β),
  (μ.prod ν) (s ×ˢ t) = μ s * ν t