theorem Th41:
  for c being convergent Complex_Sequence holds Re c is convergent
  & Im c is convergent & lim Re c = Re lim c & lim Im c = Im lim c
proof
  let c be convergent Complex_Sequence;
  ( for n holds Re c.n=Re(c.n))& for n holds Im c.n=Im(c.n) by Def5,Def6;
  hence thesis by Th40;
end;
