∀ {R : Type u_3} {M : Type u_4} [inst : CommRing R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Submodule R M}
  (s₀ : Finset ↑N.associatedPrimes),
  IsLowerSet ↑s₀ →
    ∀ (q : Submodule R M),
      q.IsPrimary →
        ∀ (p : ↑N.associatedPrimes),
          (q.colon Set.univ).radical = ↑p →
            Submodule.comap (LocalizedModule.mkLinearMap (⨅ q ∈ s₀, (↑q).primeCompl) M)
                (Submodule.localized₀ (⨅ q ∈ s₀, (↑q).primeCompl)
                  (LocalizedModule.mkLinearMap (⨅ q ∈ s₀, (↑q).primeCompl) M) q) =
              if p ∈ s₀ then q else ⊤