reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, x be Point of X, r be Real, V be Subset of
LinearTopSpaceNorm X st V = {y where y is Point of X:||.x-y.|| < r} holds V is
  open
proof
  let X be RealNormSpace, x be Point of X, r be Real, V be Subset of
  LinearTopSpaceNorm X;
  reconsider V0 = V as Subset of TopSpaceNorm X by Def4;
  assume V = {y where y is Point of X:||.x-y.|| < r};
  then V0 is open by Th8;
  hence thesis by Th20;
end;
