reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;
reserve fi,psi for Ordinal-Sequence;

theorem
  for A ex B,C st B is limit_ordinal & C is natural & A = B +^ C
proof
  defpred Th[Ordinal] means
  ex A1,A2 st A1 is limit_ordinal & A2 is natural & $1 = A1 +^ A2;
A1: for A st for B st B in A holds Th[B] holds Th[A]
  proof
    let A such that
A2: for B st B in A holds Th[B];
A3: (ex B st A = succ B) implies Th[A]
    proof
      given B such that
A4:   A = succ B;
      consider C,D such that
A5:   C is limit_ordinal and
A6:   D is natural and
A7:   B = C +^ D by A2,A4,ORDINAL1:6;
      take C, E = succ D;
      thus C is limit_ordinal by A5;
      thus E in omega by A6,ORDINAL1:def 12;
      thus thesis by A4,A7,Th28;
    end;
    (for B holds A <> succ B) implies Th[A]
    proof
      assume
A8:   for D holds A <> succ D;
      take B = A, C = 0;
      thus B is limit_ordinal by A8,ORDINAL1:29;
      thus C in omega by ORDINAL1:def 11;
      thus thesis by Th27;
    end;
    hence thesis by A3;
  end;
  thus for A holds Th[A] from ORDINAL1:sch 2(A1);
end;
