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 Th30:
    X = {} iff Ideal_X = [#]Polynom-Ring(n,R)
    proof
A1:   X = {} implies Ideal_X = [#]Polynom-Ring(n,R)
      proof
        assume
A2:     X = {};
        o in [#]Polynom-Ring(n,R) implies o in Ideal_X
        proof
          assume o in [#]Polynom-Ring(n,R); then
          reconsider f = o as Polynomial of n,R by POLYNOM1:def 11;
          {} c= Zero_(f) by XBOOLE_1:2;
          hence thesis by A2;
        end; then
        [#]Polynom-Ring(n,R) c= Ideal_X;
        hence thesis;
      end;
      Ideal_X = [#]Polynom-Ring(n,R) implies X = {}Funcs(n,[#]R)
      proof
        assume
A4:     Ideal_X = [#]Polynom-Ring(n,R);
        assume X <> {}Funcs(n,[#]R); then
        consider o such that
A6:     o in X by XBOOLE_0:def 1;
        reconsider x = o as Element of Funcs(n,[#]R) by A6;
A7:     Ideal_{x} = [#]Polynom-Ring(n,R) by A4,Th29,A6,ZFMISC_1:31;
        reconsider y = x as Function of n,R;
        1.Polynom-Ring(n,R) in [#]Polynom-Ring(n,R) by SUBSET_1:def 1;
          then
        1_(n,R) in {f where f is Polynomial of n,R : {x} c= Zero_(f)}
          by A7, POLYNOM1:31; then
        consider g be Polynomial of n,R such that
A9:     g = 1_(n,R) & {x} c= Zero_(g);
        {x} c= {}Funcs(n,[#]R) by A9,Th14;
        hence contradiction;
      end;
      hence thesis by A1;
    end;
