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

theorem Th16:
  for A being non empty set for O being Ordinal for T being
  Sequence holds O <> 0 & O is limit_ordinal & dom T = O & (for O1 being
Ordinal st O1 in O holds T.O1 = ConsecutiveSet2(A,O1)) implies ConsecutiveSet2(
  A,O) = union rng T
proof
  deffunc D(set,Sequence) = union rng $2;
  deffunc C(Ordinal,set) = new_set2 $2;
  let A be non empty set;
  let O be Ordinal;
  let T be Sequence;
  deffunc F(Ordinal) = ConsecutiveSet2(A,$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 = A & (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 Def5;
  thus F(O) = D(O,T) from ORDINAL2:sch 10(A4,A1,A2,A3);
end;
