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

theorem Th19:
  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
  ConsecutiveDelta2(q,succ O) = new_bi_fun2(BiFun(ConsecutiveDelta2(q,O),
  ConsecutiveSet2(A,O),L),Quadr2(q,O))
proof
  deffunc D(set,Sequence) = union rng $2;
  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;
  deffunc C(Ordinal,set) = new_bi_fun2(BiFun($2,ConsecutiveSet2(A,$1),L),
  Quadr2(q,$1));
  deffunc F(Ordinal) = ConsecutiveDelta2(q,$1);
A1: for O being Ordinal, It being object holds It = F(O) iff ex L0 being
Sequence st It = last L0 & dom L0 = succ O & L0.0 = d & (for C being Ordinal
st succ C in succ O holds L0.succ C = C(C,L0.C)) & for C being Ordinal st C in
  succ O & C <> 0 & C is limit_ordinal holds L0.C = D(C,L0|C) by Def7;
  for O being Ordinal holds F(succ O) = C(O,F(O)) from ORDINAL2:sch 9( A1);
  hence thesis;
end;
