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

theorem Th42:
  0_No = Unique_No 0_No
proof
  set B=born_eq 0_No,C=Unique_No 0_No;
  C in (unique_No_op B).B by Def10;
  then A1: born C c= B by Th38;
  B c= born 0_No ={} by Def5,SURREAL0:37;
  then born C ={} by A1;
  hence thesis by SURREAL0:37;
end;
