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 Th66:
  A is limit_ordinal implies
    ex X be set st No_Ordinal_op A = [X,{}] &
       (for o holds o in X iff ex B st B in A & o = No_Ordinal_op B)
proof
  assume
A1: A is limit_ordinal;
  set B=succ A;
  consider S be Sequence such that
A2:  No_Ordinal_op A = S.A & dom S = B and
A3:  (for O st succ O in B holds S.succ O = [{S.O},{}]) &
  for O st O in B & O is limit_ordinal holds S.O = [rng (S|O),{}] by Def11;
  take X= rng (S|A);
  thus No_Ordinal_op A = [X,{}] by ORDINAL1:6,A1,A2,A3;
  let o;
A4: A in B by ORDINAL1:6;
  then
A5:dom (S|A)=A by A2,ORDINAL1:def 2,RELAT_1:62;
  thus o in X implies ex B st B in A & o = No_Ordinal_op B
  proof
    assume o in X;
    then consider x be object such that
A6: x in dom (S|A) & (S|A).x =o by FUNCT_1:def 3;
    reconsider x as Ordinal by A6;
    S.x = No_Ordinal_op x by A2,A3,Th63,A4,A6,ORDINAL1:10;
    hence thesis by A6,FUNCT_1:47;
  end;
  given C be Ordinal such that
A7:C in A & o = No_Ordinal_op C;
  No_Ordinal_op C = S.C = (S|A).C
    by A7,A4,ORDINAL1:10,A5,A2,A3,Th63,FUNCT_1:47;
  hence thesis by A5,A7,FUNCT_1:def 3;
end;
