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

theorem Th23:
  for A being non empty set for L be lower-bounded LATTICE for d
  be BiFunction of A,L for q being QuadrSeq of d for O being Ordinal holds d c=
  ConsecutiveDelta2(q,O)
proof
  let A be non empty set;
  let L be lower-bounded LATTICE;
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  let O be Ordinal;
  defpred X[Ordinal] means d c= ConsecutiveDelta2(q,$1);
A1: for O1 being Ordinal st X[O1] holds X[succ O1]
  proof
    let O1 be Ordinal;
    ConsecutiveDelta2(q,succ O1) = new_bi_fun2(BiFun(ConsecutiveDelta2( q,
    O1 ) , ConsecutiveSet2(A,O1),L),Quadr2(q,O1)) by Th19
      .= new_bi_fun2(ConsecutiveDelta2(q,O1),Quadr2(q,O1)) by LATTICE5:def 15;
    then
A2: ConsecutiveDelta2(q,O1) c= ConsecutiveDelta2(q,succ O1) by Th13;
    assume d c= ConsecutiveDelta2(q,O1);
    hence thesis by A2,XBOOLE_1:1;
  end;
A3: 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) = ConsecutiveDelta2(q,$1);
    let O2 be Ordinal;
    assume that
A4: O2 <> 0 and
A5: O2 is limit_ordinal and
    for O1 be Ordinal st O1 in O2 holds d c= ConsecutiveDelta2(q,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.{} = ConsecutiveDelta2(q,{}) by A4,A7,ORDINAL3:8
      .= d by Th18;
    then
A8: d in rng Ls by A7,A6,FUNCT_1:def 3;
    ConsecutiveDelta2(q,O2) = union rng Ls by A4,A5,A7,Th20;
    hence thesis by A8,ZFMISC_1:74;
  end;
A9: X[0] by Th18;
  for O being Ordinal holds X[O] from ORDINAL2:sch 1(A9,A1,A3);
  hence thesis;
end;
