reserve X for RealNormSpace;

theorem Th14:
  for X be RealNormSpace, S be sequence of X, St be sequence of
TopSpaceNorm X st S = St & St is convergent holds Lim St = {lim S} & lim St=lim
  S
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of TopSpaceNorm X;
  assume that
A1: S = St and
A2: St is convergent;
  consider x being Point of TopSpaceNorm X such that
A3: Lim St = {x} by A2,FRECHET2:24;
  reconsider Sm=St as sequence of MetricSpaceNorm X;
A4: Sm is convergent by A2,FRECHET2:29;
  then
A5: lim St=lim Sm by FRECHET2:30
    .=lim S by A1,A4,Th6;
  x in Lim St by A3,TARSKI:def 1;
  then St is_convergent_to x by FRECHET:def 5;
  hence Lim St = {lim S} by A5,A3,FRECHET2:25;
  thus thesis by A5;
end;
