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

theorem
  born x is finite implies L_x is finite & R_x is finite
proof
  assume born x is finite;
  then A1: Day born x is finite by Th7;
  A2: L_x \/ R_x c= Day born x
  proof
    let o;
    assume A3:o in L_x \/ R_x;
    then reconsider y=o as Surreal by SURREAL0:def 16;
    born y in born x by A3,Th1;
    then y in Day born y c= Day born x by SURREAL0:35,def 18,ORDINAL1:def 2;
    hence thesis;
  end;
  L_x c= L_x \/ R_x & R_x c= L_x \/ R_x by XBOOLE_1:7;
  hence thesis by A2,A1;
end;
