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
    {0.Polynom-Ring(n,R)} c= Ideal_([#]Funcs(n,[#]R))
    proof
      o in {0.Polynom-Ring(n,R)} implies o in Ideal_([#]Funcs(n,[#]R))
      proof
        assume o in {0.Polynom-Ring(n,R)}; then
A2:     o = 0.Polynom-Ring(n,R) by TARSKI:def 1
        .= 0_(n,R) by POLYNOM1:def 11;
        Zero_(0_(n,R)) = Funcs(n,[#]R) by Th13;
        hence thesis by A2;
      end;
      hence thesis;
    end;
