∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set R),
  Ideal.span s = ⊤ →
    ∀ (Rₚ : ↑s → Type u_4) [inst_3 : (r : ↑s) → CommSemiring (Rₚ r)] [inst_4 : (r : ↑s) → Algebra R (Rₚ r)]
      [inst_5 : ∀ (r : ↑s), IsLocalization.Away (↑r) (Rₚ r)] (Mₚ : ↑s → Type u_5)
      [inst_6 : (r : ↑s) → AddCommMonoid (Mₚ r)] [inst_7 : (r : ↑s) → Module R (Mₚ r)]
      [inst_8 : (r : ↑s) → Module (Rₚ r) (Mₚ r)] [inst_9 : ∀ (r : ↑s), IsScalarTower R (Rₚ r) (Mₚ r)]
      (f : (r : ↑s) → M →ₗ[R] Mₚ r) [inst_10 : ∀ (r : ↑s), IsLocalizedModule (Submonoid.powers ↑r) (f r)]
      {N P : Submodule R M},
      (∀ (r : ↑s),
          Submodule.localized' (Rₚ r) (Submonoid.powers ↑r) (f r) N =
            Submodule.localized' (Rₚ r) (Submonoid.powers ↑r) (f r) P) →
        N = P