
theorem
for R being preordered Ring,
    P being Preordering of R
for a being P-ordered Element of R holds a is non P-positive iff a <=P, 0.R
proof
let R be preordered Ring, P be Preordering of R;
let a be P-ordered Element of R;
H: a in P \/ -P & P /\ -P = {0.R} by REALALG1:def 14,defppp;
hereby assume a is non P-positive;
   then per cases by XBOOLE_0:def 5;
   suppose not a in P;
     then a in -P by H,XBOOLE_0:def 3;
     then -a in --P;
     hence a <=P, 0.R;
     end;
   suppose a in {0.R};
     then a = 0.R by TARSKI:def 1;
     hence a <=P, 0.R by c1;
     end;
   end;
assume a <=P, 0.R;
  then C: --a in -P;
  per cases;
  suppose a = 0.R;
    then a in {0.R} by TARSKI:def 1;
    hence a is non P-positive by XBOOLE_0:def 5;
    end;
  suppose D: a <> 0.R;
    now assume a in P;
      then a in {0.R} by C,H;
      hence contradiction by D,TARSKI:def 1;
      end;
    hence a is non P-positive by XBOOLE_0:def 5;
    end;
end;
