∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} {S : Type u_2} [inst_1 : CommSemiring S]
  [inst_2 : Algebra R S] {P : Type u_3} [inst_3 : CommSemiring P] [IsLocalization M S] [inst_5 : Algebra R P]
  [IsLocalization M P] {x y : R}, (algebraMap R S) x = (algebraMap R S) y ↔ (algebraMap R P) x = (algebraMap R P) y