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 Th18:
    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\/J) c= Zero_(I)/\Zero_(J)
      proof
        Zero_(I\/J) c= Zero_(J) & Zero_(I\/J) c= Zero_(I) by XBOOLE_1:7,Th16;
        hence thesis by XBOOLE_1:19;
      end;
      Zero_(I) /\ Zero_(J) c= Zero_(I \/ J)
      proof
        for o holds o in Zero_(I) /\ Zero_(J) implies o in Zero_(I \/ J)
        proof
          let o;
          assume o in Zero_(I) /\ Zero_(J); then
A3:       o in Zero_(I) & o in Zero_(J) by XBOOLE_0:def 4; then
          o in {x where x is Function of n,R : for f be Polynomial of n,R
          st f in J holds eval(f,x) = 0.R} by Def6; then
          consider x being Function of n,R such that
A4:       o = x & for f be Polynomial of n,R holds
          f in J implies eval(f,x) = 0.R;
          o in {x where x is Function of n,R : for f be Polynomial of n,R
          st f in I holds eval(f,x) = 0.R} by A3, Def6; then
          consider y being Function of n,R such that
A5:       o = y &
          for g be Polynomial of n,R holds
          g in I implies eval(g,y) = 0.R;
          o in Zero_(I \/ J)
          proof
            assume not o in Zero_(I \/ J); then
A7:         not o in {x where x is Function of n,R :
            for f be Polynomial of n,R
            st f in I\/J holds eval(f,x) = 0.R} by Def6;
            consider f be Polynomial of n,R such that
A8:         f in I \/ J & eval(f,x) <> 0.R by A4,A7;
            per cases by A8,XBOOLE_0:def 3;
              suppose f in J;
                hence contradiction by A8,A4;
              end;
              suppose f in I;
                hence contradiction by A8,A4,A5;
              end;
            end;
            hence thesis;
          end;
          hence thesis;
        end;
        hence thesis by A1;
      end;
