∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompletePartialOrderSup α]
  [inst_1 : ConditionallyCompletePartialOrderSup β] {l : α → β} {u : β → α},
  GaloisConnection l u →
    ∀ {s : Set α}, DirectedOn (fun x1 x2 => x1 ≤ x2) s → s.Nonempty → BddAbove s → l (sSup s) = ⨆ x, l ↑x