reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th64:
  No_Ordinal_op 0 = 0_No
proof
  consider S be Sequence such that
A1:  No_Ordinal_op 0 = S.0 & dom S = succ 0 and
A2:  (for O st succ O in succ 0 holds S.succ O = [{S.O},{}]) &
  for O st O in succ 0 & O is limit_ordinal holds
  S.O = [rng (S|O),{}] by Def11;
  S.0 = [rng (S|0),{}] by A2,ORDINAL1:6,ORDINAL2:4;
  hence thesis by A1;
end;
