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

theorem Th44:
  x <= y & z < t implies x+z < y+t
proof
  assume A1: x <= y & z < t;
  assume not x+z < y+t;
  then A2: z >= y+t - x by Th42;
  A3: x-x ==0_No by Th39;
  x+-x <= y +-x by A1,Th32;
  then 0_No <= y +-x by A3,SURREALO:4;
  then A4: t+0_No <= t+(y +-x) by Th32;
  y+t +- x = t+(y +- x) by Th37;
  hence thesis by A1,A2,A4,SURREALO:4;
end;
