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

theorem Th41:
  x+y <= z iff x <= z - y
proof
  thus x+y <= z implies x <= z - y
  proof
    assume x+y <= z;
    then A1: x+y +-y <= z +-y by Th32;
    y -y == 0_No by Th39;
    then x+(y +-y) >= x+0_No by Th32;
    then x+y +-y >= x by Th37;
    hence thesis by A1,SURREALO:4;
  end;
  assume 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,SURREALO:4;
end;
