reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th46:
  b <> {} & b is limit_ordinal implies
  for phi being Ordinal-Sequence st dom phi = b &
     for c st c in b holds phi.c = epsilon_c
  holds epsilon_b = Union phi
  proof assume
A1: b <> {} & b is limit_ordinal;
    let f; assume
A2: dom f = b & for c st c in b holds f.c = epsilon_c; then
    f is increasing Ordinal-Sequence by Th45; then
    Union f is_limes_of f by A1,A2,Th6; then
    Union f = lim f by ORDINAL2:def 10;
    hence thesis by A1,A2,Th43;
  end;
