∀ {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α}
  (motive : (x : ℕ) → (∀ i < x, m.IsCaratheodory (s i)) → Prop) (x : ℕ) (x_1 : ∀ i < x, m.IsCaratheodory (s i)),
  (∀ (x : ∀ i < 0, m.IsCaratheodory (s i)), motive 0 x) →
    (∀ (n : ℕ) (h : ∀ i < n + 1, m.IsCaratheodory (s i)), motive n.succ h) → motive x x_1