reserve x for set,
  C for Ordinal,
  L0 for Sequence;
reserve O1,O2 for Ordinal;

theorem Th17:
  for A being non empty set for O being Ordinal holds A c= ConsecutiveSet2(A,O)
proof
  let A be non empty set;
  let O be Ordinal;
  defpred X[Ordinal] means A c= ConsecutiveSet2(A,$1);
A1: for O1 being Ordinal st X[O1] holds X[succ O1]
  proof
    let O1 be Ordinal;
    ConsecutiveSet2(A,succ O1) = new_set2 ConsecutiveSet2(A,O1) by Th15;
    then
A2: ConsecutiveSet2(A,O1) c= ConsecutiveSet2(A,succ O1) by XBOOLE_1:7;
    assume A c= ConsecutiveSet2(A,O1);
    hence thesis by A2,XBOOLE_1:1;
  end;
A3: for O1 st O1 <> 0 & O1 is limit_ordinal & for O2 st O2 in O1 holds X[O2
  ] holds X[O1]
  proof
    deffunc U(Ordinal) = ConsecutiveSet2(A,$1);
    let O2 be Ordinal;
    assume that
A4: O2 <> 0 and
A5: O2 is limit_ordinal and
    for O1 be Ordinal st O1 in O2 holds A c= ConsecutiveSet2(A,O1);
A6: {} in O2 by A4,ORDINAL3:8;
    consider Ls being Sequence such that
A7: dom Ls = O2 & for O1 being Ordinal st O1 in O2 holds Ls.O1 = U(O1)
    from ORDINAL2:sch 2;
    Ls.{} = ConsecutiveSet2(A,{}) by A4,A7,ORDINAL3:8
      .= A by Th14;
    then
A8: A in rng Ls by A7,A6,FUNCT_1:def 3;
    ConsecutiveSet2(A,O2) = union rng Ls by A4,A5,A7,Th16;
    hence thesis by A8,ZFMISC_1:74;
  end;
A9: X[0] by Th14;
  for O being Ordinal holds X[O] from ORDINAL2:sch 1(A9,A1,A3);
  hence thesis;
end;
