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 Th16:
    for S,T be non empty Subset of Polynom-Ring(n,R) holds
    S c= T implies Zero_(T) c= Zero_(S)
    proof
      let S,T be non empty Subset of Polynom-Ring(n,R);
      assume
A1:   S c= T;
      for o holds o in Zero_(T) implies o in Zero_(S)
      proof
        let o;
        assume o in Zero_(T); then
        o in {x where x is Function of n,R : for f be Polynomial of n,R
        st f in T holds eval(f,x) = 0.R} by Def6; then
        consider x being Function of n,R such that
A3:     o = x & for f be Polynomial of n,R st f in T holds
        eval(f,x) = 0.R;
        for f be Polynomial of n,R holds f in S implies eval(f,x) = 0.R
        by A1,A3; then
        x 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};
        hence thesis by A3,Def6;
      end;
      hence thesis;
    end;
