reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, S be sequence of X, St be sequence of
  LinearTopSpaceNorm X st S=St holds St is convergent iff S is convergent
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of
  LinearTopSpaceNorm X;
  assume
A1: S=St;
  the carrier of LinearTopSpaceNorm X = the carrier of TopSpaceNorm X by Def4;
  then reconsider SSt = St as sequence of TopSpaceNorm X;
  St is convergent iff SSt is convergent by Th27;
  hence thesis by A1,Th13;
end;
