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 Th34:
    Zero_(Ideal_(Zero_S)) = Zero_S
    proof
      per cases;
        suppose S <> {}; then
      reconsider S as non empty Subset of Polynom-Ring(n,R);
          Zero_S c= Zero_(Ideal_(Zero_S)) by Th33;
          hence thesis by Th16,Th32;
        end;
        suppose
A3:       S = {}; then
          Zero_(Ideal_(Zero_S)) = Zero_([#]Polynom-Ring(n,R)) by Def6, Th30
          .= {}Funcs(n,[#]R) by Lm4 .= Zero_S by Def6,A3;
          hence thesis;
        end;
      end;
