reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th37:
  |.x + y.| <= |.x.| + |.y.|
proof
  - |.x.| <= x <= |.x.| & - |.y.| <= y <= |.y.| by Th34;
  then - (|.x.| +|.y.|) = - |.x.| + -|.y.| <= x+y <= |.x.| +|.y.|
  by SURREALR:40,43;
  hence thesis by Th35;
end;
