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

theorem Th21:
  for A being non empty set for O,O1,O2 being Ordinal holds O1 c=
  O2 implies ConsecutiveSet2(A,O1) c= ConsecutiveSet2(A,O2)
proof
  let A be non empty set;
  let O,O1,02 be Ordinal;
  defpred X[Ordinal] means O1 c= $1 implies ConsecutiveSet2(A,O1) c=
  ConsecutiveSet2(A,$1);
A1: for O1 being Ordinal st X[O1] holds X[succ O1]
  proof
    let O2 be Ordinal;
    assume
A2: O1 c= O2 implies ConsecutiveSet2(A,O1) c= ConsecutiveSet2(A,O2);
    assume
A3: O1 c= succ O2;
    per cases;
    suppose
      O1 = succ O2;
      hence thesis;
    end;
    suppose
      O1 <> succ O2;
      then O1 c< succ O2 by A3;
      then
A4:   O1 in succ O2 by ORDINAL1:11;
      ConsecutiveSet2(A,O2) c= new_set2 ConsecutiveSet2(A,O2) by XBOOLE_1:7;
      then ConsecutiveSet2(A,O1) c= new_set2 ConsecutiveSet2(A,O2) by A2,A4,
ORDINAL1:22;
      hence thesis by Th15;
    end;
  end;
A5: 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
A6: O2 <> 0 & O2 is limit_ordinal and
    for O3 being Ordinal st O3 in O2 holds O1 c= O3 implies
    ConsecutiveSet2(A,O1) c= ConsecutiveSet2(A,O3);
    consider L being Sequence such that
A7: dom L = O2 & for O3 being Ordinal st O3 in O2 holds L.O3 = U(O3)
    from ORDINAL2:sch 2;
A8: ConsecutiveSet2(A,O2) = union rng L by A6,A7,Th16;
    assume
A9: O1 c= O2;
    per cases;
    suppose
      O1 = O2;
      hence thesis;
    end;
    suppose
      O1 <> O2;
      then
A10:  O1 c< O2 by A9;
      then O1 in O2 by ORDINAL1:11;
      then
A11:  L.O1 in rng L by A7,FUNCT_1:def 3;
      L.O1 = ConsecutiveSet2(A,O1) by A7,A10,ORDINAL1:11;
      hence thesis by A8,A11,ZFMISC_1:74;
    end;
  end;
A12: X[0];
  for O being Ordinal holds X[O] from ORDINAL2:sch 1(A12,A1,A5);
  hence thesis;
end;
