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 Th65:
  No_Ordinal_op succ A = [{No_Ordinal_op A},{}]
proof
  set B=succ A;
  consider S be Sequence such that
A1:  No_Ordinal_op B = S.B & dom S = succ B and
A2:  (for O st succ O in succ B holds S.succ O = [{S.O},{}]) &
  for O st O in succ B & O is limit_ordinal holds S.O = [rng (S|O),{}]
  by Def11;
A3:B in succ B & A in B by ORDINAL1:6;
  S.B = [{S.A},{}] by ORDINAL1:6,A2;
  hence thesis by A3,A1,A2,Th63,ORDINAL1:10;
end;
