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 Th33:
    X c= Zero_(Ideal_X)
    proof
      for o holds o in X implies o in Zero_(Ideal_X)
      proof
        let o;
        assume
A1:     o in X; then
     reconsider X as non empty Subset of Funcs(n,[#]R) by XBOOLE_0:def 1;
        consider x be Function such that
A2:     o = x  & dom x = n & rng x c= [#]R by A1,FUNCT_2:def 2;
    reconsider x as Function of n,R by A2,FUNCT_2:2;
        o in Zero_(Ideal_X)
        proof
          assume not o in Zero_(Ideal_X); then
          not o in {z where z is Function of n,R :for f be Polynomial of n,R
          st f in Ideal_X holds eval(f,z) = 0.R} by Def6; then
          consider f1 be Polynomial of n,R such that
A4:       f1 in Ideal_X & eval(f1,x) <> 0.R by A2;
          consider f2 be Polynomial of n,R such that
A5:       f1 = f2 & X c= Zero_(f2) by A4;
          o in Zero_(f1) by A1,A5; then
          consider x1 be Function of n,R such that
A6:       x1 = o & eval(f1,x1) = 0.R;
          thus contradiction by A2,A4,A6;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
