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

theorem Th12:
  -y <= x & x <= y iff |.x.| <= y
proof
A1: -y <= x & x <= y implies |.x.| <= y
  proof
    assume that
A2: -y <= x and
A3: x <= y;
    per cases;
    suppose
      0 <= x;
      hence thesis by A3,Def1;
    end;
    suppose
A4:   not 0 <= x;
      -x <= y by A2,XXREAL_3:60;
      hence thesis by A4,Def1;
    end;
  end;
  |.x.| <= y implies -y <= x & x <= y
  proof
    assume
A5: |.x.| <= y;
    per cases by A5,XXREAL_0:1;
    suppose
      |.x.| = y;
      hence thesis by Th9;
    end;
    suppose
      |.x.| < y;
      hence thesis by Th10;
    end;
  end;
  hence thesis by A1;
end;
