theorem
  I c= J implies sqrt I c= sqrt J
  proof
    assume
A1: I c= J;
    let s be object;
    assume
A2: s in sqrt I; then
    reconsider s as Element of A;
    s in {a where a is Element of A:
    ex n being Element of NAT st a|^n in I } by A2,IDEAL_1:def 24; then
    consider s1 be Element of A such that
A3: s1 = s and
A4: ex n being Element of NAT st s1|^n in I;
    consider n1 be Element of NAT such that
A5: s1|^n1 in I by A4;
    n1 <> 0
    proof
      assume n1 = 0; then
      s1|^n1 = 1_A by BINOM:8;
      hence contradiction by A1,A5,IDEAL_1:19;
    end; then
    reconsider n1 as non zero Nat;
    s1 in {a where a is Element of A:ex n being Element of NAT st a|^n in J}
      by A1,A5;
    hence thesis by A3,IDEAL_1:def 24;
  end;
