
theorem Th17:
  for X be RealNormSpace, x be sequence of X
  holds RAT_Sums(rng x) is Subset of the carrier of NLin(rng x)
  proof
    let X be RealNormSpace, x be sequence of X;
    set D = RAT_Sums(rng x);
    for z be object st z in D holds z in the carrier of NLin(rng x)
    proof
      let z be object;
      assume z in D; then
      consider l be  Linear_Combination of rng x such that
      A2: z = Sum l & rng l c= RAT;
      z in Lin(rng x) by A2,RLVECT_3:14;
      hence z in the carrier of NLin(rng x);
    end;
    hence thesis by TARSKI:def 3;
  end;
