∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {P : Type u_3}
  [inst_2 : CommMonoid P] (f : S.LocalizationMap N) {g : M →* P},
  (∀ (y : ↥S), IsUnit (g ↑y)) → ∀ {x y : M}, f x = f y → g x = g y