reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem
  born x = {} iff x = 0_No
proof
  hereby assume born x={};
    then x in Day {} by Def18;
    hence x=0_No by Th2,TARSKI:def 1;
  end;
  assume x = 0_No;
  then x in Games {} by Th2,TARSKI:def 1;
  then x in Day {} by Th8;
  then born x c= {} by Def18;
  hence thesis;
end;
