
theorem
  for L being non degenerated comRing, p, q being Polynomial of L holds
  Roots p \/ Roots q c= Roots (p*'q)
proof
  let L be non degenerated comRing, p, q being Polynomial of L;
  let x be object;
  assume
A1: x in Roots p \/ Roots q;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: x in Roots p;
    then reconsider a = x as Element of L;
    a is_a_root_of p by A2,POLYNOM5:def 10;
    then eval(p,a) = 0.L;
    then eval(p,a) * eval(q,a) = 0.L;
    then eval(p*'q,a) = 0.L by POLYNOM4:24;
    then a is_a_root_of p*'q;
    hence thesis by POLYNOM5:def 10;
  end;
  suppose
A3: x in Roots q;
    then reconsider a = x as Element of L;
    a is_a_root_of q by A3,POLYNOM5:def 10;
    then eval(q,a) = 0.L;
    then eval(p,a) * eval(q,a) = 0.L;
    then eval(p*'q,a) = 0.L by POLYNOM4:24;
    then a is_a_root_of p*'q;
    hence thesis by POLYNOM5:def 10;
  end;
end;
