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
     sqrt I = sqrt(sqrt I)
     proof
       for o be object st o in sqrt(sqrt I) holds o in sqrt I
       proof
         let o be object;
         assume
A1:      o in sqrt(sqrt I); 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 (sqrt I)} by A1,IDEAL_1:def 24; then
         consider o1 be Element of A such that
A2:      o1 = o and
A3:      ex n being Element of NAT st o1|^n in (sqrt I);
         consider n1 be Element of NAT such that
A4:      o1|^n1 in (sqrt I) by A3;
       reconsider x = o1|^n1 as Element of A;
         x in {a where a is Element of A: ex n being Element of NAT st
               a|^n in I} by A4,IDEAL_1:def 24; then
         consider x1 be Element of A such that
A5:      x1 = x and
A6:      ex m being Element of NAT st x1|^m in I;
         consider m1 be Element of NAT such that
A7:      x1|^m1 in I by A6;
         reconsider nm = n1*m1 as Element of NAT;
         o1|^nm in I by A5,A7,BINOM:11; then
         o1 in {a where a is Element of A: ex n being Element of NAT st
                a|^n in I};
         hence thesis by A2,IDEAL_1:def 24;
       end; then
       sqrt(sqrt I) c= sqrt I;
       hence thesis by TOPZARI1:20;
     end;
