reserve X for RealNormSpace;

theorem Th27:
  for X be RealNormSpace, S be sequence of TopSpaceNorm 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 TopSpaceNorm X, St be sequence of
  LinearTopSpaceNorm X;
  assume
A1: S=St;
A2: now
    assume S is convergent;
    then consider x be Point of TopSpaceNorm X such that
A3: S is_convergent_to x by FRECHET:def 4;
    reconsider xt=x as Point of LinearTopSpaceNorm X by Def4;
    St is_convergent_to xt by A1,A3,Th26;
    hence St is convergent by FRECHET:def 4;
  end;
  now
    assume St is convergent;
    then consider xt be Point of LinearTopSpaceNorm X such that
A4: St is_convergent_to xt by FRECHET:def 4;
    reconsider x=xt as Point of TopSpaceNorm X by Def4;
    S is_convergent_to x by A1,A4,Th26;
    hence S is convergent by FRECHET:def 4;
  end;
  hence thesis by A2;
end;
