∀ {α : Type u_1} [inst : LinearOrder α] {s : Set α} {a : α} (u : ℕ → α) (n : ℕ),
  StrictMonoOn u (Set.Iic (n + 1)) →
    (∀ i ∈ Set.Iic (n + 1), u i ∈ s ∩ Set.Iic a) →
      u (n + 1) ≤ a →
        u (n + 1) = a →
          ∀ b ∈ s ∩ Set.Ioo (u n) (u (n + 1)), ∀ i ≤ n + 1, (fun i => if i ≤ n then u i else b) i ∈ s ∩ Set.Iio a