reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;

theorem
  for f being Sequence st f is descending & f.0 is finite &
  for a st f.a <> {} holds succ a in dom f
  holds last f = {}
  proof
    let f be Sequence such that
A1: f is descending & f.0 is finite and
A2: for a st f.a <> {} holds succ a in dom f;
    f is finite by A1,Th27; then
    reconsider d = dom f as finite Ordinal;
    set u = union d;
    per cases;
    suppose d = 0;
      hence last f = {} by FUNCT_1:def 2;
    end;
    suppose d <> 0; then
      consider n being Nat such that
A3:   d = n+1 by NAT_1:6;
      d = Segm(n+1) by A3;
      then
A4:   d = succ Segm n by NAT_1:38; then
A5:   u = n by ORDINAL2:2;
      f.u <> 0 implies d in d by A2,A4,A5;
      hence last f = {};
    end;
  end;
