reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, x be Point of X, r be Real, V be Subset of
  TopSpaceNorm X st V = {y where y is Point of X:||.x-y.|| <= r} holds V is
  closed
proof
  let X be RealNormSpace, x be Point of X, r be Real, V be Subset of
  TopSpaceNorm X;
  assume
A1: V = {y where y is Point of X:||.x-y.|| <= r};
  reconsider z=x as Element of MetricSpaceNorm X;
  ex t be Point of X st t=x & cl_Ball(z,r) = {y where y is Point of X: ||.
  t-y.|| <= r} by Th3;
  hence thesis by A1,TOPREAL6:57;
end;
