reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;

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;
