
theorem SSF:
  for f be real-valued XFinSequence holds Sum f = Sum Sequel f
  proof
    let f be real-valued XFinSequence;
    reconsider g = Re Sequel f as summable Real_Sequence;
    reconsider n = len f as Nat;
    A2: Sum (Sequel f) = Sum g + Sum (Im (Sequel f))*<i> by COMSEQ_3:53
    .= Sum g + 0*<i>
    .= Sum g;
    per cases;
    suppose n = 0; then
      f is empty; then
      Sum f = 0 & Sum Sequel f = 0;
      hence thesis;
    end;
    suppose n > 0; then
      reconsider k = n-1 as Nat;
      Sum g = (Partial_Sums g).k + Sum (g^\(k+1)) by SERIES_1:15
      .= (Partial_Sums g).k + 0
      .= Sum (g|(k+1)) by AFINSQ_2:56
      .= Sum f;
      hence thesis by A2;
    end;
  end;
