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 Th31:
  for d be BiFunction of A,L for q being QuadrSeq of d holds d c=
  ConsecutiveDelta(q,O)
proof
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  defpred X[Ordinal] means d c= ConsecutiveDelta(q,$1);
A1: 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) = ConsecutiveDelta(q,$1);
    let O2;
    assume that
A2: O2 <> 0 and
A3: O2 is limit_ordinal and
    for O1 st O1 in O2 holds d c= ConsecutiveDelta(q,O1);
A4: {} in O2 by A2,ORDINAL3:8;
    consider Ls being Sequence such that
A5: dom Ls = O2 & for O1 being Ordinal st O1 in O2 holds Ls.O1 = U(O1)
    from ORDINAL2:sch 2;
    Ls.{} = ConsecutiveDelta(q,{}) by A2,A5,ORDINAL3:8
      .= d by Th26;
    then
A6: d in rng Ls by A5,A4,FUNCT_1:def 3;
    ConsecutiveDelta(q,O2) = union rng Ls by A2,A3,A5,Th28;
    hence thesis by A6,ZFMISC_1:74;
  end;
A7: for O1 st X[O1] holds X[succ O1]
  proof
    let O1;
    ConsecutiveDelta(q,succ O1) = new_bi_fun(BiFun(ConsecutiveDelta(q,O1),
    ConsecutiveSet(A,O1),L),Quadr(q,O1)) by Th27
      .= new_bi_fun(ConsecutiveDelta(q,O1),Quadr(q,O1)) by Def15;
    then
A8: ConsecutiveDelta(q,O1) c= ConsecutiveDelta(q,succ O1) by Th19;
    assume d c= ConsecutiveDelta(q,O1);
    hence thesis by A8,XBOOLE_1:1;
  end;
A9: X[0] by Th26;
  for O holds X[O] from ORDINAL2:sch 1(A9,A7,A1);
  hence thesis;
end;
