
theorem Th19:
  for X be RealNormSpace, x be sequence of X
  holds RAT_Sums(rng x) is countable Subset of the carrier of ClNLin(rng x)
  proof
    let X be RealNormSpace, x be sequence of X;
    set D = RAT_Sums(rng x);
    A1: D is Subset of the carrier of NLin(rng x) by Th17;
    the carrier of NLin(rng x) c= the carrier of ClNLin(rng x) by Th18;
    hence thesis by A1,XBOOLE_1:1;
  end;
