reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, V be Subset of 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 X, Vt be Subset of LinearTopSpaceNorm
  X;
  reconsider VVt = Vt as Subset of TopSpaceNorm X by Def4;
  assume
A1: V=Vt;
  Vt is closed iff VVt is closed by Th21;
  hence thesis by A1,Th15;
end;
