 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

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;
