
theorem av00:
for R being preordered non degenerated Ring,
    P being Preordering of R,
    a being Element of R holds not a is P-ordered iff abs(P,a) = -1.R
proof
let R be preordered non degenerated Ring, P be Preordering of R,
    a be Element of R;
  hereby assume not a is P-ordered;
    then not(a in P) & not(a in -P) by XBOOLE_0:def 3;
    hence abs(P,a) = -1.R by defa;
  end;
  assume AS: abs(P,a) = -1.R;
   assume a is P-ordered;
   then 0.R <=P, abs(P,a) by av0;
   hence contradiction by AS,REALALG1:26;
end;
