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 Th21:
    for f,g be Polynomial of n,R holds Zero_({f*'g}) = Zero_({f}) \/ Zero_({g})
    proof
      let f,g be Polynomial of n,R;
A1:   o in Zero_({f*'g}) implies o in Zero_({f}) \/ Zero_({g})
      proof
        assume o in Zero_({f*'g}); then
        o in Zero_(f*'g) by Th15; then
        consider x be Function of n,R such that
A4:     x = o & eval(f*'g,x) = 0.R;
        reconsider x1 = eval(f,x) as Element of R;
        reconsider x2 = eval(g,x) as Element of R;
A5:     0.R = x1 * x2 by A4,POLYNOM2:25;
     :::Integral Domain requires
        eval(f,x) = 0.R or eval(g,x) = 0.R by A5,VECTSP_2:def 1; then
        o in Zero_(f) or o in Zero_(g) by A4; then
        o in Zero_({f}) or o in Zero_({g}) by Th15;
        hence thesis by XBOOLE_0:def 3;
      end;
      o in Zero_({f}) \/ Zero_({g}) implies o in Zero_({f*'g})
      proof
        assume o in Zero_({f}) \/ Zero_({g}); then
        o in Zero_({f}) or o in Zero_({g}) by XBOOLE_0:def 3; then
        per cases by Th15;
          suppose o in Zero_(f); then
            consider x be Function of n,R such that
A10:        o = x & eval(f,x) = 0.R;
            reconsider x1 = eval(g,x) as Element of R;
            eval(f*'g,x) = 0.R * x1 by A10,POLYNOM2:25 .= 0.R; then
            o in Zero_(f*'g) by A10;
            hence thesis by Th15;
          end;
          suppose o in Zero_(g); then
            consider x be Function of n,R such that
A13:        o = x & eval(g,x) = 0.R;
            reconsider x1 = eval(f,x) as Element of R;
            eval(f*'g,x) = x1 * 0.R by A13,POLYNOM2:25 .= 0.R; then
            o in Zero_(f*'g) by A13;
            hence thesis by Th15;
          end;
        end;
        hence thesis by A1,TARSKI_0:2;
      end;
