∀ {α : Type u_2} {β : Type u_1} [inst : LinearOrder β] [inst_1 : SuccOrder β] {s : β → Set α},
  (∀ (n : β), (s n ∩ s (Order.succ n)).Nonempty) →
    ∀ (x x_1 i : β), i ∈ Set.Ico x_1 x → (s (Order.succ i) ∩ s i).Nonempty