∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {k : α} {f : β → β},
  (t.modify k f).isEmpty = false →
    ∀ {fallback : α},
      (t.modify k f).maxKeyD fallback = if cmp (t.maxKeyD fallback) k = Ordering.eq then k else t.maxKeyD fallback