reserve X for RealNormSpace;

theorem Th21:
  for X be RealNormSpace, V be Subset of TopSpaceNorm X, Vt be
  Subset of LinearTopSpaceNorm X st V=Vt holds V is closed iff Vt is closed
proof
  let X be RealNormSpace, V be Subset of TopSpaceNorm X, Vt be Subset of
  LinearTopSpaceNorm X;
  assume V=Vt;
  then
A1: Vt` =V` by Def4;
  Vt is closed iff V` is open by A1,Th20;
  hence thesis;
end;
