
theorem
  for f be complex-valued XFinSequence holds Sum (f|1) = f.0
  proof
    let f be complex-valued XFinSequence;
    per cases;
    suppose
      not f is empty; then
      1 <= len f by NAT_1:14; then
      B2: len (f|1) = 1 by AFINSQ_1:54; then
      0 in Segm (len (f|1)) by NAT_1:44; then
      0 in dom f & 0 in 1 by RELAT_1:57; then
      B3: 0 in (dom f) /\ 1 by XBOOLE_0:def 4;
      Sum (f|1) = Sum <%(f|1).0%> by B2,AFINSQ_1:34
      .= f.0 by B3,FUNCT_1:48;
      hence thesis;
    end;
    suppose
      f is empty;
      hence thesis;
    end;
  end;
