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

theorem
  L_x <> {x} <> R_x
proof
  A1:x in {x} by TARSKI:def 1;
  thus L_x <> {x}
  proof
    assume L_x = {x};
    then x in L_x \/R_x by A1,XBOOLE_0:def 3;
    then born x in born x by Th1;
    hence thesis;
  end;
  assume R_x = {x};
  then x in L_x \/R_x by A1,XBOOLE_0:def 3;
  then born x in born x by Th1;
  hence thesis;
end;
