reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th24:
  A c= ConsecutiveSet(A,O)
proof
  defpred X[Ordinal] means A c= ConsecutiveSet(A,$1);
A1: for O1 st X[O1] holds X[succ O1]
  proof
    let O1;
    ConsecutiveSet(A,succ O1) = new_set ConsecutiveSet(A,O1) by Th22;
    then
A2: ConsecutiveSet(A,O1) c= ConsecutiveSet(A,succ O1) by XBOOLE_1:7;
    assume A c= ConsecutiveSet(A,O1);
    hence thesis by A2,XBOOLE_1:1;
  end;
A3: for O2 st O2 <> 0 & O2 is limit_ordinal & for O1 st O1 in O2 holds X[O1
  ] holds X[O2]
  proof
    deffunc U(Ordinal) = ConsecutiveSet(A,$1);
    let O2;
    assume that
A4: O2 <> 0 and
A5: O2 is limit_ordinal and
    for O1 st O1 in O2 holds A c= ConsecutiveSet(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.{} = ConsecutiveSet(A,{}) by A4,A7,ORDINAL3:8
      .= A by Th21;
    then
A8: A in rng Ls by A7,A6,FUNCT_1:def 3;
    ConsecutiveSet(A,O2) = union rng Ls by A4,A5,A7,Th23;
    hence thesis by A8,ZFMISC_1:74;
  end;
A9: X[0] by Th21;
  for O holds X[O] from ORDINAL2:sch 1(A9,A1,A3);
  hence thesis;
end;
