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;

theorem
  for f being array holds f is 0-limited iff f is empty
  proof let f be array;
    thus f is 0-limited implies f is empty
    proof assume sup dom f = 0; then
      dom f c= 0 by ORDINAL6:3;
      hence f is empty;
    end;
    assume f is empty;
    hence sup dom f = 0 by ORDINAL2:18;
  end;
