reserve n for Nat;

theorem
ex p,q being Polynomial of Z/6 st not Roots(p*'q) c= Roots(p) \/ Roots(q)
proof
set R = Z/6, z = 2 '*' 1.(Z/6), d = 3 '*' 1.(Z/6);
take p = rpoly(1,z), q = rpoly(1,d);
C: Char R = 6 by RING_3:77;
eval(p*'q,0.R) = eval(p,0.R) * eval(q,0.R) by POLYNOM4:24
              .= (0.R - z) * eval(q,0.R) by HURWITZ:29
              .= (0.R - z) * (0.R - d) by HURWITZ:29
              .= (-z) * (0.R - d) by RLVECT_1:14
              .= (-z) * (- d) by RLVECT_1:14
              .= z * d by VECTSP_1:10
              .= (2 * 3) '*' 1.R by RING_3:67
              .= 0.R by C,RING_3:def 5;
then 0.R is_a_root_of (p*'q) by POLYNOM5:def 7;
then B: 0.R in Roots(p*'q) by POLYNOM5:def 10;
now assume AS: 0.R in Roots(p) \/ Roots(q);
  per cases by AS,XBOOLE_0:def 3;
  suppose 0.R in Roots(p);
    then 0.R is_a_root_of p by POLYNOM5:def 10;
    then B: 0.R = eval(p,0.R) by POLYNOM5:def 7
               .= 0.R - z by HURWITZ:29
               .= - z by RLVECT_1:14;
    z = --z .= 0.R by B;
    hence contradiction by C,RING_3:def 5;
    end;
  suppose 0.R in Roots(q);
    then 0.R is_a_root_of q by POLYNOM5:def 10;
    then B: 0.R = eval(q,0.R) by POLYNOM5:def 7
               .= 0.R - d by HURWITZ:29
               .= - d by RLVECT_1:14;
    d = --d .= 0.R by B;
    hence contradiction by C,RING_3:def 5;
    end;
  end;
hence thesis by B;
end;
