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 Th32:
    S c= Ideal_(Zero_(S))
    proof
      for o holds o in S implies o in Ideal_(Zero_(S))
      proof
        let o;
        assume
A1:     o in S; then
A2:     S <> {} by XBOOLE_0:def 1;
        reconsider p = o as Polynomial of n,R by A1,POLYNOM1:def 11;
        assume not o in Ideal_(Zero_(S)); then
        not Zero_(S) c= Zero_(p); then
        consider z be object such that
A4:     z in Zero_(S) and
A5:     not z in Zero_(p);
        z in {x where x is Function of n,R :
        for f be Polynomial of n,R st f in S holds eval(f,x) = 0.R}
        by A2,A4,Def6; then
        consider x be Function of n,R such that
A6:     x = z and
A7:     for f be Polynomial of n,R st f in S holds eval(f,x) = 0.R;
        eval(p,x) = 0.R by A1,A7;
        hence contradiction by A5, A6;
      end;
      hence thesis;
    end;
