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 Th29:
  O1 c= O2 implies ConsecutiveSet(A,O1) c= ConsecutiveSet(A,O2)
proof
  defpred X[Ordinal] means O1 c= $1 implies ConsecutiveSet(A,O1) c=
  ConsecutiveSet(A,$1);
A1: for O2 st X[O2] holds X[succ O2]
  proof
    let O2;
    assume
A2: O1 c= O2 implies ConsecutiveSet(A,O1) c= ConsecutiveSet(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;
      ConsecutiveSet(A,O2) c= new_set ConsecutiveSet(A,O2) by XBOOLE_1:7;
      then ConsecutiveSet(A,O1) c= new_set ConsecutiveSet(A,O2) by A2,A4,
ORDINAL1:22;
      hence thesis by Th22;
    end;
  end;
A5: for O2 st O2 <> 0 & O2 is limit_ordinal & for O3 st O3 in O2 holds X[
  O3] holds X[O2]
  proof
    deffunc U(Ordinal) = ConsecutiveSet(A,$1);
    let O2;
    assume that
A6: O2 <> 0 & O2 is limit_ordinal and
    for O3 st O3 in O2 holds O1 c= O3 implies ConsecutiveSet(A,O1) c=
    ConsecutiveSet(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: ConsecutiveSet(A,O2) = union rng L by A6,A7,Th23;
    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 = ConsecutiveSet(A,O1) by A7,A10,ORDINAL1:11;
      hence thesis by A8,A11,ZFMISC_1:74;
    end;
  end;
A12: X[0];
  for O2 holds X[O2] from ORDINAL2:sch 1(A12,A1,A5);
  hence thesis;
end;
