
theorem
  for M be non empty MetrSpace,
      V be Subset of TopSpaceMetr M st V is open holds
    ex F be Subset-Family of M
      st F = {Ball(x,r) where x is Element of M, r is Real:
              r > 0 & Ball(x,r) c= V}
       & V = union F
  proof
    let M be non empty MetrSpace,
        V be Subset of TopSpaceMetr M;
    assume
    A1: V is open;
    set F = {Ball(x,r) where x is Element of M, r is Real:
             r > 0 & Ball(x,r) c= V};
    for z be object st z in F holds z in Family_open_set M
    proof
      let z be object;
      assume z in F; then
      ex x be Element of M, r be Real
        st z = Ball(x,r) & r > 0 & Ball(x,r) c= V;
      hence z in Family_open_set M by PCOMPS_1:29;
    end; then
    F c= Family_open_set M by TARSKI:def 3; then
    reconsider F as Subset-Family of M by XBOOLE_1:1;
    take F;
    thus F = {Ball(x,r) where x is Element of M, r is Real:
              r > 0 & Ball(x,r) c= V};
    reconsider Q = union F as Subset of TopSpaceMetr M;
    for z be object holds z in V iff z in Q
    proof
      let z be object;
      hereby
        assume
        A2: z in V; then
        reconsider x = z as Element of M;
        consider r be Real such that
        A3: r > 0 & Ball(x,r) c= V by A1,A2,TOPMETR:15;
        dist(x,x) = 0 by METRIC_1:1; then
        A4: x in Ball(x,r) by A3,METRIC_1:11;
        Ball(x,r) in F by A3;
        hence z in Q by A4,TARSKI:def 4;
      end;
      assume z in Q; then
      consider B be set such that
      A5: z in B & B in F by TARSKI:def 4;
      consider x be Element of M, r be Real such that
      A6: B = Ball(x,r) & r > 0 & Ball(x,r) c= V by A5;
      thus z in V by A5,A6;
    end;
    hence thesis by TARSKI:2;
  end;
