∀ {α : Type u_2} {β : Type u_3} [inst : Preorder α] [inst_1 : Preorder β] {l : α → β} {u : β → α}
  (choice : (x : α) → u (l x) ≤ x → β) (gc : GaloisConnection l u) (le_l_u : ∀ (x : β), x ≤ l (u x))
  (choice_eq : ∀ (a : α) (h : u (l a) ≤ a), choice a h = l a) (choice_1 : (x : α) → u (l x) ≤ x → β)
  (gc_1 : GaloisConnection l u) (le_l_u_1 : ∀ (x : β), x ≤ l (u x))
  (choice_eq_1 : ∀ (a : α) (h : u (l a) ≤ a), choice_1 a h = l a),
  ({ choice := choice, gc := gc, le_l_u := le_l_u, choice_eq := choice_eq } =
      { choice := choice_1, gc := gc_1, le_l_u := le_l_u_1, choice_eq := choice_eq_1 }) =
    (choice = choice_1)