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 Th33:
    I is primary iff I <> [#]A &
    for x,y be Element of A st x*y in I & not x in I holds y in sqrt I
    proof
      (I <> [#]A &
      for a1,a2 be Element of A st a1*a2 in I & not a1 in I holds a2 in sqrt I)
      implies I is primary
      proof
        assume
A1:     I <> [#]A &
        for a1,a2 be Element of A st a1*a2 in I & not a1 in I
        holds a2 in sqrt I; then
A2:     I is quasi-primary;
        I is proper by A1;
        hence thesis by A2;
      end;
      hence thesis by Def4;
    end;
