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 Th22:
    for I,J be Ideal of Polynom-Ring(n,R) holds
    Zero_(I /\ J) = Zero_(I) \/ Zero_(J)
    proof
      let I,J be Ideal of Polynom-Ring(n,R);
A1:   Zero_I c= Zero_(I /\ J) & Zero_J c= Zero_(I /\ J) by Th16,XBOOLE_1:17;
      for o holds o in Zero_(I /\ J) implies o in (Zero_(I) \/ Zero_(J))
      proof
        let o;
        assume o in Zero_(I /\ J); then
        o in {x where x is Function of n,R : for p be Polynomial of n,R
              st p in I /\ J holds eval(p,x) = 0.R} by Def6; then
        consider x1 be Function of n,R such that
A3:     o = x1 &
        for p be Polynomial of n,R st p in I /\ J holds eval(p,x1) = 0.R;
        x1 in (Zero_(I) \/ Zero_(J))
        proof
          assume not x1 in (Zero_(I) \/ Zero_(J)); then
A5:       not x1 in Zero_(I) & not x1 in Zero_(J) by XBOOLE_0:def 3;
          not x1 in  {z where z is Function of n,R: for f be Polynomial of n,R
          st f in I holds eval(f,z) = 0.R} by A5,Def6; then
          consider f1 be Polynomial of n,R such that
A6:       f1 in I & eval(f1,x1) <> 0.R;
          not x1 in {z where z is Function of n,R: for f be Polynomial of n,R
          st f in J holds eval(f,z) = 0.R} by A5,Def6; then
          consider f2 be Polynomial of n,R such that
A7:       f2 in J & eval(f2,x1) <> 0.R;
A8:       eval(f1*'f2,x1) = eval(f1,x1)*eval(f2,x1) by POLYNOM2:25;
      reconsider F1 = f1, F2 = f2 as Element of Polynom-Ring(n,R)
      by POLYNOM1:def 11;
A9:       F1*F2 = f1*'f2 by POLYNOM1:def 11;
A10:      F1*F2 in I & F1*F2 in J by A6,A7,IDEAL_1:def 2;
          f1*'f2 in I/\J by A9,A10;
          hence contradiction by A3,A8,A6,A7,VECTSP_2:def 1;
        end;
        hence thesis by A3;
      end; then
      Zero_(I /\ J) c= Zero_(I) \/ Zero_(J);
      hence thesis by A1,XBOOLE_1:8;
    end;
