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

theorem Th13:
  |.x + y.| <= |.x.| + |.y.|
proof
A1: -|.x.| <= x & -|.y.| <= y by Th9;
A2: x <= |.x.| & y <= |.y.| by Th9;
A3:     -|.x.|-|.y.| = -(|.x.|+|.y.|) by XXREAL_3:25;
  x + y <= |.x.| + |.y.| & -|.x.|+(-|.y.|) <= x + y by A1,A2,XXREAL_3:36;
  hence thesis by A3,Th12;
end;
