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

theorem
  |.x.| <= z & |.y.| <= w implies |.x+y.| <= z + w
proof
  assume
A1: |.x.| <= z & |.y.| <= w;
  then -z <= x & -w <= y by Th12;
  then
A2: -z + -w <= x + y by XXREAL_3:36;
  x <= z & y <= w by A1,Th12;
  then
A3: x + y <= z + w by XXREAL_3:36;
  -z -w = -(z + w) by XXREAL_3:25;
  hence thesis by A3,A2,Th12;
end;
