reserve x,y,w,z for ExtReal,
  a for Real;

theorem Th9:
  -|.x.| <= x & x <= |.x.|
proof
  reconsider x as R_eal by XXREAL_0:def 1;
  per cases;
  suppose
A1: 0 <= x;
    thus thesis by A1,Def1;
  end;
  suppose
 not 0 <= x;
    then |.x.|=-x by Def1;
    hence thesis;
  end;
end;
