
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 p <> 0_.(L) & z is_a_root_of p holds deg(p div rpoly(1,z)) =
  deg(p) - 1
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;
  assume that
A1: p <> 0_.(L) and
A2: z is_a_root_of p;
  consider s being Polynomial of L such that
A3: p = rpoly(1,z) *' s by A2,Th33;
A4: rpoly(1,z) <> 0_.(L) by A1,A3,POLYNOM4:2;
A5: ex t being Polynomial of L st p = s *' rpoly(1,z) + t & deg t < deg
  rpoly(1,z)
  proof
    take t = 0_.(L);
    thus s *' rpoly(1,z) + t = p by A3,POLYNOM3:28;
    deg t = -1 by Th20;
    hence thesis by Th27;
  end;
  s <> 0_.(L) by A1,A3,POLYNOM3:34;
  then deg p = deg(rpoly(1,z)) + deg(s) by A3,A4,Th23
    .= 1 + deg(s) by Th27;
  hence thesis by A4,A5,Def5;
end;
