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 Th13:
    sqrt I = [#]A iff I = [#]A
    proof
::     =>)
      thus sqrt I = [#]A implies I = [#]A
      proof
        assume sqrt I = [#]A; then
        1.A in sqrt I; then
        1.A in {a where a is Element of A: ex n being Element of NAT st
                a|^n in I} by IDEAL_1:def 24; then
        consider o1 be Element of A such that
A3:     o1 = 1.A and
A4:     ex n being Element of NAT st o1|^n in I;
        consider m1 be Element of NAT such that
A5:     o1|^m1 in I by A4;
A6:     o1|^m1 = 1.A by A3,TOPZARI1:2;
        not (I is proper) by A5,A6,IDEAL_1:19;
        hence thesis;
      end;
::     <=)
      thus thesis by TOPZARI1:20;
    end;
