
theorem
for R being preordered non degenerated Ring
for P being Preordering of R
holds Positives(Poly) P <> LowPositives(Poly) P
proof
let R be preordered non degenerated Ring, P be Preordering of R;
set p = rpoly(1,1.R);
reconsider cp = {i where i is Nat : p.i <> 0.R} as non empty Subset of NAT
  by lemlp1;
deg p = 1 by HURWITZ:27;
then len p -' 1 = 1 by XREAL_0:def 2;
then p.(len p-'1) = 1_R by HURWITZ:25;
then B: LC p = 1.R;
A: 1.R in P by ord3;
E: p.0 = -(power(R).(1.R,0+1)) by HURWITZ:25
      .= -((power(R).(1.R,0)) * 1.R) by GROUP_1:def 7
      .= -(1_R * 1.R) by GROUP_1:def 7
      .= - 1.R;
now assume - 1.R = 0.R;
  then -(-1.R) = 0.R;
  hence contradiction;
  end;
then 0 in cp by E;
then C: min* cp = 0 by NAT_1:def 1;
now assume p in LowPositives(Poly) P;
  then consider q being Polynomial of R such that
  H: q = p & q.(min*{i where i is Nat : q.i <> 0.R}) in P;
  thus contradiction by H,C,E,ord4;
  end;
hence thesis by A,B;
end;
