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 Th23:
  O <> 0 & O is limit_ordinal & dom T = O & (for O1 being Ordinal
  st O1 in O holds T.O1 = ConsecutiveSet(A,O1)) implies ConsecutiveSet(A,O) =
  union rng T
proof
  deffunc V(Ordinal,Sequence) = union rng $2;
  deffunc U(Ordinal,set) = new_set $2;
  deffunc F(Ordinal) = ConsecutiveSet(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, x being object holds x = F(O) iff ex L0 being
Sequence st x = last L0 & dom L0 = succ O & L0.0 = A & (for C being Ordinal
st succ C in succ O holds L0.succ C = U(C,L0.C)) & for C being Ordinal st C in
  succ O & C <> 0 & C is limit_ordinal holds L0.C = V(C,L0|C) by Def12;
  thus F(O) = V(O,T) from ORDINAL2:sch 10(A4,A1,A2,A3);
end;
