∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : ConditionallyCompletePartialOrderSup α]
  [inst_1 : ConditionallyCompletePartialOrderSup β] {l : α → β} {u : β → α},
  GaloisConnection l u →
    ∀ {s : Set γ} {f : γ → α},
      DirectedOn (fun x1 x2 => x1 ≤ x2) (f '' s) → BddAbove (f '' s) → s.Nonempty → l (⨆ i, f ↑i) = ⨆ i, l (f ↑i)