reserve R for commutative Ring;
reserve A for non degenerated commutative Ring;
reserve I,J,q for Ideal of A;
reserve p for prime Ideal of A;
reserve M,M1,M2 for Ideal of A/q;

theorem Th45:
    for Q be proper Ideal of A holds
    (for x,y be Element of A holds x*y in Q & not y in sqrt Q implies x in Q)
    implies
    Q is primary & sqrt Q is prime
    proof
      let Q be proper Ideal of A;
      assume
A3:   for x,y be Element of A
         holds x*y in Q & not y in sqrt Q implies x in Q;
      not 1.A in sqrt Q by TOPZARI1:3; then
      sqrt Q is proper; then
      reconsider P = sqrt Q as proper Ideal of A;
      reconsider Q as proper Ideal of A;
A2:   Q c= P & P c= sqrt Q by TOPZARI1:20;
      for x,y be Element of A holds x*y in Q & not(x in Q) implies y in P
        by A3;
      hence thesis by A2,Th44;
    end;
