reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem
    sqrt(Ideal_X) = Ideal_X
    proof
      per cases;
        suppose X <> {}; then
          Ideal_X <> [#]Polynom-Ring(n,R) by Th30; then
reconsider IX = Ideal_X as proper Ideal of Polynom-Ring(n,R)
             by SUBSET_1:def 6;
A4:       not 1.Polynom-Ring(n,R) in IX by IDEAL_1:19;
          sqrt(Ideal_X) c= Ideal_X
          proof
            now let o;
              assume o in sqrt(Ideal_X); then
              consider f be Element of Polynom-Ring(n,R) such that
A5:           f = o & ex m be Element of NAT st f|^m in Ideal_X;
              consider m be Element of NAT such that
A6:           f|^m in Ideal_X by A5;
              m <> 0
              proof
                assume
A7:             m = 0;
                f|^m = (power Polynom-Ring(n,R)).(f,m) by BINOM:def 2
                .= 1_Polynom-Ring(n,R) by A7,GROUP_1:def 7
                .= 1.Polynom-Ring(n,R);
                hence contradiction by A4,A6;
              end; then
              reconsider m as non zero Nat;
              f in [#]Polynom-Ring(n,R) by SUBSET_1:def 1; then
              reconsider p = f as Polynomial of n,R by POLYNOM1:def 11;
              consider fm be Polynomial of n,R such that
A9:           fm = f|^m & X c= Zero_(fm) by A6;
              f|^m = p`^m by BINOM:def 2; then
              Zero_(fm) = Zero_({p`^m}) by A9,Th15 .= Zero_({p}) by Lm8
              .= Zero_(p) by Th15;
              hence o in Ideal_X by A5,A9;
            end;
            hence thesis;
          end;
          hence thesis by TOPZARI1:20;
        end;
        suppose X = {}; then
          Ideal_X = [#]Polynom-Ring(n,R) by Th30;
          hence thesis by IDEAL_2:10;
        end;
      end;
