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

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