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 Th29:
    X c= Y implies Ideal_Y c= Ideal_X
    proof
      assume
A1:   X c= Y;
      o in Ideal_Y implies o in Ideal_X
      proof
        assume o in Ideal_Y; then
        consider f be Polynomial of n,R such that
A3:     f = o & Y c= Zero_(f);
        X c= Zero_(f) by A3,A1;
        hence thesis by A3;
      end;
      hence thesis;
    end;
