theorem
  for x being Point of TopSpaceMetr(Euclid 1),
      y being Point of Euclid 1,
      cB being basis of BOOL2F NeighborhoodSystem x,
      b being Element of cB st x = y & cB = Balls(x) holds
  ex n being Nat
  st b = {q where q is Element of Euclid 1: dist(y,q) < 1/n}
  proof
    let x be Point of TopSpaceMetr(Euclid 1),
        y be Point of Euclid 1,
        cB be basis of BOOL2F NeighborhoodSystem x,
        b be Element of cB;
    assume that
A1: x = y and
A2: cB = Balls(x);
    consider z be Point of Euclid 1 such that
A3: x = z and
A4: Balls(x) = {Ball(z,1/n) where n is Nat : n <> 0 } by FRECHET:def 1;
    b in {Ball(z,1/n) where n is Nat : n <> 0 } by A2,A4;
    then consider n be Nat such that
A5: b = Ball(z,1/n) and n <> 0;
    Ball(y,1/n) = {q where q is Element of Euclid 1: dist(y,q) < 1/n}
      by METRIC_1:def 14;
    hence thesis by A5,A1,A3;
  end;
