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 Th32:
  for d being BiFunction of A,L for q being QuadrSeq of d st O1 c=
  O2 holds ConsecutiveDelta(q,O1) c= ConsecutiveDelta(q,O2)
proof
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  defpred X[Ordinal] means O1 c= $1 implies ConsecutiveDelta(q,O1) c=
  ConsecutiveDelta(q,$1);
A1: 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) = ConsecutiveDelta(q,$1);
    let O2;
    assume that
A2: O2 <> 0 & O2 is limit_ordinal and
    for O3 st O3 in O2 holds O1 c= O3 implies ConsecutiveDelta(q,O1) c=
    ConsecutiveDelta(q,O3);
    consider L being Sequence such that
A3: dom L = O2 & for O3 being Ordinal st O3 in O2 holds L.O3 = U(O3)
    from ORDINAL2:sch 2;
A4: ConsecutiveDelta(q,O2) = union rng L by A2,A3,Th28;
    assume
A5: O1 c= O2;
    per cases;
    suppose
      O1 = O2;
      hence thesis;
    end;
    suppose
      O1 <> O2;
      then
A6:   O1 c< O2 by A5;
      then O1 in O2 by ORDINAL1:11;
      then
A7:   L.O1 in rng L by A3,FUNCT_1:def 3;
      L.O1 = ConsecutiveDelta(q,O1) by A3,A6,ORDINAL1:11;
      hence thesis by A4,A7,ZFMISC_1:74;
    end;
  end;
A8: for O2 st X[O2] holds X[succ O2]
  proof
    let O2;
    assume
A9: O1 c= O2 implies ConsecutiveDelta(q,O1) c= ConsecutiveDelta(q,O2);
    assume
A10: O1 c= succ O2;
    per cases;
    suppose
      O1 = succ O2;
      hence thesis;
    end;
    suppose
      O1 <> succ O2;
      then O1 c< succ O2 by A10;
      then
A11:  O1 in succ O2 by ORDINAL1:11;
      ConsecutiveDelta(q,succ O2) = new_bi_fun(BiFun(ConsecutiveDelta(q,O2
      ), ConsecutiveSet(A,O2),L),Quadr(q,O2)) by Th27
        .= new_bi_fun(ConsecutiveDelta(q,O2),Quadr(q,O2)) by Def15;
      then ConsecutiveDelta(q,O2) c= ConsecutiveDelta(q,succ O2) by Th19;
      hence thesis by A9,A11,ORDINAL1:22;
    end;
  end;
A12: X[0];
  for O2 holds X[O2] from ORDINAL2:sch 1(A12,A8,A1);
  hence thesis;
end;
