reserve X for RealNormSpace;

theorem Th28:
  for X be RealNormSpace, S be sequence of TopSpaceNorm X, St be
  sequence of LinearTopSpaceNorm X st S=St & St is convergent holds Lim S = Lim
  St & lim S = lim St
proof
  let X be RealNormSpace, S be sequence of TopSpaceNorm X, St be sequence of
  LinearTopSpaceNorm X;
  assume that
A1: S=St and
A2: St is convergent;
A3: S is convergent by A1,A2,Th27;
  then consider x be Point of TopSpaceNorm X such that
A4: S is_convergent_to x by FRECHET:def 4;
  reconsider xxt=x as Point of LinearTopSpaceNorm X by Def4;
  St is_convergent_to xxt by A1,A4,Th26;
  then
A5: lim St = xxt by FRECHET2:25
    .=lim S by A4,FRECHET2:25;
  reconsider Sn=S as sequence of X;
  consider xt be Point of LinearTopSpaceNorm X such that
A6: Lim St = {xt} by A2,FRECHET2:24;
  xt in {xt} by TARSKI:def 1;
  then St is_convergent_to xt by A6,FRECHET:def 5;
  then Lim St = {lim St} by A6,FRECHET2:25
    .= {lim Sn} by A3,A5,Th14
    .= Lim S by A3,Th14;
  hence thesis by A5;
end;
