∀ {R : Type u_5} {S : Type u_6} {Sₘ : Type u_7} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing Sₘ]
  [inst_3 : Algebra R S] [inst_4 : Algebra S Sₘ] [inst_5 : Algebra R Sₘ] [IsScalarTower R S Sₘ] (M : Submonoid R)
  [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] {x : S},
  IsIntegral R ((algebraMap S Sₘ) x) → ∃ m ∈ M, IsIntegral R (m • x)