reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th47:
  A <> {} implies ex C,D st B = C*^A+^D & D in A
proof
  defpred I[Ordinal] means ex C,D st $1 = C*^A+^D & D in A;
  assume
A1: A <> {};
A2: for B st B <> 0 & B is limit_ordinal & for A1 st A1 in B holds I[A1]
  holds I[B]
  proof
    {} in A by A1,Th8;
    then
A3: succ 0 c= A by ORDINAL1:21;
    let B such that
    B <> 0 and
A4: B is limit_ordinal and
    for A1 st A1 in B holds I[A1];
    defpred P[Ordinal] means $1 in B & B in $1*^A;
    B*^1 = B by ORDINAL2:39;
    then
A5: B c= B*^A by A3,ORDINAL2:42;
A6: now
      assume B <> B*^A;
      then B c< B*^A by A5;
      then B in B*^A by ORDINAL1:11;
      then
A7:   ex C st P[C] by A4,Th41;
      consider C such that
A8:   P[C] and
A9:   for C1 being Ordinal st P[C1] holds C c= C1 from ORDINAL1:sch 1
      (A7);
      now
        assume C is limit_ordinal;
        then consider C1 being Ordinal such that
A10:    C1 in C and
A11:    B in C1*^A by A8,Th41;
        C1 in B by A8,A10,ORDINAL1:10;
        hence contradiction by A9,A10,A11,ORDINAL1:5;
      end;
      then consider C1 being Ordinal such that
A12:  C = succ C1 by ORDINAL1:29;
A13:  C1 in C by A12,ORDINAL1:6;
      then C1 in B by A8,ORDINAL1:10;
      then not B in C1*^A by A9,A13,ORDINAL1:5;
      then consider D such that
A14:  B = C1*^A+^D by Th27,ORDINAL1:16;
      thus I[B]
      proof
        take C1,D;
        thus B = C1*^A+^D by A14;
        C1*^A+^D in C1*^A+^A by A8,A12,A14,ORDINAL2:36;
        hence thesis by Th22;
      end;
    end;
    B = B*^A implies B = B*^A+^{} & {} in A by A1,Th8,ORDINAL2:27;
    hence thesis by A6;
  end;
A15: for B st I[B] holds I[succ B]
  proof
    let B;
    given C,D such that
A16: B = C*^A+^D and
A17: D in A;
A18: now
      assume not succ D in A;
      then
A19:  A c= succ D by ORDINAL1:16;
      take C1 = succ C, D1 = {};
      succ D c= A by A17,ORDINAL1:21;
      then
A20:  A = succ D by A19;
      thus C1*^A+^D1 = C1*^A by ORDINAL2:27
        .= C*^A+^A by ORDINAL2:36
        .= succ B by A16,A20,ORDINAL2:28;
      thus D1 in A by A1,Th8;
    end;
    now
      assume
A21:  succ D in A;
      take C1 = C, D1 = succ D;
      thus C1*^A+^D1 = succ B by A16,ORDINAL2:28;
      thus D1 in A by A21;
    end;
    hence thesis by A18;
  end;
A22: I[0]
  proof
    take C = {}, D = {};
    thus 0 = {}+^{} by ORDINAL2:27
      .= C*^A+^D by ORDINAL2:35;
    thus thesis by A1,Th8;
  end;
  for B holds I[B] from ORDINAL2:sch 1(A22,A15,A2);
  hence thesis;
end;
