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
    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);
set PR = Polynom-Ring(n,R);
A1:   Zero_(I/\J) c= Zero_(I*'J) by Th16,IDEAL_1:79;
      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; then
          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 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;
   reconsider F1=f1,F2=f2 as Element of Polynom-Ring(n,R) by POLYNOM1:def 11;
A8:       F1*F2 =f1*'f2 by POLYNOM1:def 11;
   reconsider s1 = <* F1*F2 *> as FinSequence of the carrier of PR;
          dom s1 = Seg 1 by FINSEQ_1:def 8; then
A10:      len s1 = 1 by FINSEQ_1:def 3;
A11:       Sum s1 = F1*F2 by BINOM:3;
          set M = { Sum s where s is FinSequence of the carrier of PR :
          for i being Element of NAT st 1 <= i & i <= len s ex a,b being
          Element of PR st s.i = a*b & a in I & b in J};
          for i being Element of NAT st 1 <= i & i <= len s1 ex a,b being
          Element of PR st s1.i = a*b & a in I & b in J
          proof
            let i be Element of NAT;
            assume 1 <= i & i <= len s1; then
A13:        i = 1 by A10,XXREAL_0:1;
            consider a,b be Element of PR such that
A14:        a = F1 & b = F2 & s1.i = a*b & a in I & b in J by A6,A7,A13;
            thus thesis by A14;
          end; then
          f1*'f2 in I*'J by A8,A11; then
          0.R = eval(f1*'f2,x1) by A3.= eval(f1,x1)*eval(f2,x1) by POLYNOM2:25;
          hence contradiction by 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,Th22;
    end;
