reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;
reserve Rseq for Function of [:NAT,NAT:],REAL;

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;
