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 Th12:
     sqrt(I /\ J) = sqrt(I) /\ sqrt(J)
     proof
A1:    for o be object st o in sqrt(I /\ J) holds o in sqrt(I) /\ sqrt(J)
       proof
         let o be object;
         assume
A2:      o in sqrt(I /\ J); then
         reconsider o as Element of A;
         o in {a where a is Element of A: ex n being Element of NAT st
               a|^n in (I /\ J)} by A2,IDEAL_1:def 24; then
         consider o1 be Element of A such that
A3:      o1 = o and
A4:      ex n being Element of NAT st o1|^n in I /\ J;
         consider n1 be Element of NAT such that
A5:      o1|^n1 in I /\ J by A4;
A6:      o1|^n1 in I & o1|^n1 in J by A5,XBOOLE_0:def 4; then
         o1 in {a where a is Element of A: ex n being Element of NAT st
                a|^n in I}; then
A7:      o1 in sqrt I by IDEAL_1:def 24;
         o1 in {a where a is Element of A: ex n being Element of NAT st
                a|^n in J} by A6; then
         o1 in sqrt J by IDEAL_1:def 24;
         hence thesis by A3,A7,XBOOLE_0:def 4;
       end;
       sqrt(I) /\ sqrt(J) c= sqrt(I /\ J)
       proof
         let o be object;
         assume
A9:      o in sqrt(I) /\ sqrt(J); then
A10:     o in sqrt(I) & o in sqrt(J) by XBOOLE_0:def 4;
         reconsider o as Element of A by A9;
         o in {a where a is Element of A: ex n being Element of NAT st
               a|^n in I} by A10,IDEAL_1:def 24; then
         consider o1 be Element of A such that
A11:     o1 = o and
A12:     ex n being Element of NAT st o1|^n in I;
         consider n1 be Element of NAT such that
A13:     o1|^n1 in I by A12;
         o in {a where a is Element of A: ex n being Element of NAT st
               a|^n in J} by A10,IDEAL_1:def 24; then
         consider o2 be Element of A such that
A14:     o2 = o and
A15:     ex m being Element of NAT st o2|^m in J;
         consider m1 be Element of NAT such that
A16:     o2|^m1 in J by A15;
A17:     (o1|^n1)*(o1|^m1) = o1|^(n1+m1) by BINOM:10;
         reconsider n2 = n1+m1 as Element of NAT;
         o1|^n2 in I & o1|^n2 in J by A16, A14,A11, IDEAL_1:def 2, A13,A17;then
         o1|^n2 in I /\ J by XBOOLE_0:def 4; then
         o1 in {a where a is Element of A: ex n being Element of NAT st
                a|^n in I /\ J};
         hence thesis by A11,IDEAL_1:def 24;
       end;
       hence thesis by A1,TARSKI:2;
     end;
