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 Th44:
    for I,J be proper Ideal of A holds
    I c= J & J c= sqrt I &
    (for x, y be Element of A holds x*y in I & not(x in I) implies y in J)
    implies
    I is primary & sqrt I = J & J is prime
    proof
      let I,J be proper Ideal of A;
      assume that
A1:   I c= J & J c= sqrt I and
A2:   for x, y be Element of A holds x*y in I & not(x in I) implies y in J;
      for x,y be Element of A st x*y in I & not (x in I) holds y in sqrt I
        by A1,A2; then
A3:   I is primary by Def4;
      sqrt I c= J
      proof
        let x be object;
        assume x in sqrt I; then
        reconsider x0 = x as Element of sqrt I;
        consider m be Nat such that
A5:       m in {n where n is Element of NAT: not(x0|^n in I)} &
           x0|^(m+1) in I by Th43;
        consider m0 be Element of NAT such that
A6:       m0 = m & not(x0|^m0 in I) by A5;
        per cases by NAT_1:25;
          suppose
            m = 0; then
            x0|^(m+1) = x0 by BINOM:8;
            hence thesis by A1,A5;
          end;
          suppose
            1 <= m;
            reconsider x0 as Element of A;
            x0|^(m+1) = (x0|^m)*(x0|^1) by BINOM:10
            .= (x0|^m)*x0 by BINOM:8;
            hence thesis by A2,A5,A6;
          end;
        end; then
        sqrt I = J by A1;
        hence thesis by A3;
      end;
