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

theorem Th20:
  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 T being Sequence for O
  being Ordinal holds O <> 0 & O is limit_ordinal & dom T = O & (for O1 being
  Ordinal st O1 in O holds T.O1 = ConsecutiveDelta2(q,O1)) implies
  ConsecutiveDelta2(q,O) = union rng T
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 T be Sequence;
  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);
  assume that
A1: O <> 0 & O is limit_ordinal and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds T.O1 = F(O1);
A4: 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;
  thus F(O) = D(O,T) from ORDINAL2:sch 10(A4,A1,A2,A3);
end;
