∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {r : Localization S → Localization S → Sort u}
  [h : ∀ (a c : M) (b d : ↥S), Subsingleton (r (Localization.mk a b) (Localization.mk c d))] (x y : Localization S)
  (f : (a c : M) → (b d : ↥S) → r (Localization.mk a b) (Localization.mk c d)),
  x.recOnSubsingleton₂ y f =
    Quotient.recOnSubsingleton₂' x y fun t =>
      Prod.rec (motive := fun x => (a₂ : M × ↥S) → r (Quotient.mk'' x) (Quotient.mk'' a₂))
        (fun x x_1 t => Prod.rec (fun x_2 x_3 => f x x_2 x_1 x_3) t) t