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

theorem Th1:
  y in L_x \/R_x implies born y in born x
proof
  assume A1: y in L_x \/R_x;
  A2:x= [L_x,R_x];
  x in Day born x by SURREAL0:def 18;
  then consider O be Ordinal such that
  A3: O in born x & y in Day O by SURREAL0:46,A1,A2;
  born y c= O by A3,SURREAL0:def 18;
  hence thesis by A3,ORDINAL1:12;
end;
