
theorem av0:
for R being preordered non degenerated Ring,
    P being Preordering of R,
    a being Element of R holds 0.R <=P, abs(P,a) iff a is P-ordered
proof
let R be preordered non degenerated Ring, O be Preordering of R,
    a be Element of R;
hereby assume 0.R <=O, abs(O,a);
  then C: -1.R <O, abs(O,a) by avb5,c20;
  per cases;
    suppose a in O;
      hence a is O-ordered by XBOOLE_0:def 3;
      end;
    suppose a in -O;
      hence a is O-ordered by XBOOLE_0:def 3;
      end;
    suppose not(a in O) & not(a in -O);
      hence a is O-ordered by C,defa;
      end;
  end;
assume a is O-ordered;
  then per cases by XBOOLE_0:def 3;
  suppose a in O;
    hence 0.R <=O, abs(O,a) by defa;
    end;
  suppose D: a in -O;
    then -a in --O;
    hence 0.R <=O, abs(O,a) by D,defa;
    end;
end;
