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

theorem
  -y <= x & x <= y iff |.x.| <= y
proof
  hereby
    assume that
A1: -y <= x and
A2: x <= y;
    -x <= --y by A1,XREAL_1:24;
    hence |.x.| <= y by A2,Def1;
  end;
  assume
A3: |.x.| <= y;
    then
A4: 0 <= y by COMPLEX1:46;
  per cases;
  suppose 0 < x;
    hence thesis by A3,A4,Def1;
  end;
  suppose
A5: x < 0;
    then -x <= y by A3,Def1;
    then -y <= --x by XREAL_1:24;
    hence thesis by A5;
  end;
  suppose x = -0;
    hence thesis by A3;
  end;
end;
