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 Th22:
  ConsecutiveSet(A,succ O) = new_set ConsecutiveSet(A,O)
proof
  deffunc V(Ordinal,Sequence) = union rng $2;
  deffunc U(Ordinal,set) = new_set $2;
  deffunc F(Ordinal) = ConsecutiveSet(A,$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 = 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;
  for O holds F(succ O) = U(O,F(O)) from ORDINAL2:sch 9(A1);
  hence thesis;
end;
