
theorem av3:
for R being preordered domRing,
    P being Preordering of R,
    a being Element of R holds abs(P,a) = -a iff a <=P, 0.R
proof
let R be preordered domRing, O be Preordering of R,
    a be Element of R;
hereby assume B: abs(O,a) = -a;
   per cases;
    suppose a in -O;
      then -a in --O;
      hence a <=O, 0.R;
      end;
    suppose a in O;
      hence a <=O, 0.R by B,defa;
      end;
    suppose D: not(a in O) & not(a in -O);
      then --a = -- 1.R by B,defa;
      hence a <=O, 0.R by D,REALALG1:25;
      end;
  end;
assume a <=O, 0.R;
  then --a in -O;
  hence abs(O,a) = -a by defa;
end;
