∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]
  (x y : A), LocalizedModule.mk (x + y) 1 = LocalizedModule.mk x 1 + LocalizedModule.mk y 1