
theorem av2:
for R being preordered domRing,
    P being Preordering of R,
    a being Element of R holds abs(P,a) = a iff 0.R <=P, a
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;
      hence 0.R <=O, a;
      end;
    suppose a in -O;
      then a = -a by B,defa;
      then a = 0.R by tA;
      hence 0.R <=O, a by c1;
      end;
    suppose D: not(a in O) & not(a in -O);
      then -a = --1.R by B,defa;
      then -a in O by REALALG1:25;
      then --a in -O;
      hence 0.R <=O, a by D;
      end;
  end;
thus thesis by defa;
end;
