theorem
  nilrad A = meet Spectrum A
  proof
    reconsider I = {0.A} as proper Ideal of A by SUBSET_1:def 6;
    nilrad A = sqrt({0.A}) by Th27 .= meet PrimeIdeals(A,I) by Th28
      .= meet Spectrum A by Th21;
    hence thesis;
  end;
