reserve x, y, z, r, s, t for Real;

theorem
  -|.x.| <= x & x <= |.x.|
proof
  per cases;
  suppose
A1: x < 0;
    then |.x.| = -x by Def1;
    hence thesis by A1;
  end;
  suppose
A2: 0 <= x;
    then -x <= -0;
    hence thesis by A2,Def1;
  end;
end;
