
theorem
  for f be complex-valued XFinSequence holds Sum f = Sum Sequel f
  proof
    let f be complex-valued XFinSequence;
    reconsider g = (Re (Sequel f)) as real-valued Complex_Sequence;
    reconsider h = Im (Sequel f) as real-valued Complex_Sequence;
    A1: Sum (Re f) = Sum (Sequel (Re f)) & Sum (Im f) = Sum (Sequel (Im f))
      by SSF;
    A2: Sequel (Re f) = Re (Sequel f) & Sequel (Im f) = Im (Sequel f)
      by RSF,ISF;
    Sum f = Sum (Sequel (Re f)) + <i>* Sum (Sequel (Im f)) by A1,RSI
    .= Sum g + Sum (<i>(#)h) by A2,COMSEQ_3:56
    .= Sum (g + <i>(#)h) by COMSEQ_3:54;
    hence thesis;
  end;
