reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem Th32:
  x <= y iff x+z <= y+z
proof
  set Bx=born x,By=born y,Bz=born z;
  set Bxz=Bx(+)Bz,Byz=By(+)Bz;
  Bxz c= Bxz\/Byz & Byz c= Bxz\/Byz by XBOOLE_1:7;
  hence thesis by Lm5;
end;
