∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) {S : Type u_2} [inst_1 : CommSemiring S]
  [inst_2 : Algebra R S] [IsLocalization M S] {P : Type u_4} [inst_4 : Monoid P] (j k : S → P),
  j 1 = 1 →
    k 1 = 1 →
      (∀ (a b : S), j (a * b) = j a * j b) →
        (∀ (a b : S), k (a * b) = k a * k b) → (∀ (a : R), j ((algebraMap R S) a) = k ((algebraMap R S) a)) → j = k