
theorem
  for N being RealNormSpace,
      x being Point of TopSpaceMetr MetricSpaceNorm N,
      y being Point of MetricSpaceNorm N,
      n being positive Nat st x=y holds
  Ball(y,1/n) in Balls x
  proof
    let N be RealNormSpace, x be Point of TopSpaceMetr MetricSpaceNorm N,
    y be Point of MetricSpaceNorm N,
    n be positive Nat such that
A1: x=y;
    set M=MetricSpaceNorm N;
    consider y1 be Point of M such that
A2: y1=x and
A3: Balls x = {Ball(y1,1/n) where n is Nat:n <> 0} by FRECHET:def 1;
    thus thesis by A1,A2,A3;
  end;
