
theorem
  for L being Abelian add-associative right_zeroed right_complementable
  well-unital associative commutative distributive almost_left_invertible non
  degenerated non empty doubleLoopStr for p being Polynomial of L for z being
  Element of L st z is_a_root_of p holds rpoly(1,z) divides p
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital associative commutative distributive almost_left_invertible non
  degenerated non empty doubleLoopStr, p be Polynomial of L;
  let z be Element of L;
  rpoly(1,z).1 = 1_L by Lm10;
  then
A1: rpoly(1,z) <> 0_.(L);
  assume z is_a_root_of p;
  then ex s being Polynomial of L st p = rpoly(1,z) *' s by Th33;
  hence thesis by A1,Th34;
end;
