reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, V be Subset of LinearTopSpaceNorm X holds V is
  open iff for x be Point of X st x in V
 ex r be Real st r>0 & {y where y
  is Point of X:||.x-y.|| < r} c= V
proof
  let X be RealNormSpace, V be Subset of LinearTopSpaceNorm X;
  reconsider V0 = V as Subset of TopSpaceNorm X by Def4;
  V is open iff V0 is open by Th20;
  hence thesis by Th7;
end;
