theorem Th42:
  a in U implies for f being Ordinal-Sequence of a,U holds sup f in U
  proof assume
A1: a in U;
    let f be Ordinal-Sequence of a,U;
    reconsider u = Union f as Ordinal of U by Th41,A1;
    On rng f = rng f by Th2; then
    sup f c= succ u by ORDINAL3:72;
    hence sup f in U by CLASSES1:def 1;
  end;
