reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, S be sequence of X, St be sequence of
LinearTopSpaceNorm 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
  LinearTopSpaceNorm X;
  assume that
A1: S=St and
A2: St is convergent;
  the carrier of LinearTopSpaceNorm X = the carrier of TopSpaceNorm X by Def4;
  then reconsider SSt = St as sequence of TopSpaceNorm X;
  SSt is convergent by A2,Th27;
  then Lim SSt = {lim S} & lim SSt=lim S by A1,Th14;
  hence thesis by A2,Th28;
end;
