
theorem RSI:
  for f be complex-valued XFinSequence holds
    Sum f = (Sum Re f) + <i>*(Sum Im f)
  proof
    let f be complex-valued XFinSequence;
    reconsider g = Re f as (len f)-element real-valued XFinSequence;
    len (<i>(#)(Im f)) = dom (Im f) by VALUED_1:def 5
    .= len f by CARD_1:def 7; then
    reconsider h = <i>(#)Im f as (len f)-element complex-valued XFinSequence
      by CARD_1:def 7;
    Sum f = Sum (Re f + <i>(#) (Im f))
    .= Sum g + Sum h by SPP
    .= Sum (Re f) + <i>*(Sum (Im f)) by AFINSQ_2:64;
    hence thesis;
  end;
