
theorem Th8:
  for M being non empty MetrSpace, P being non empty Subset of
  TopSpaceMetr M, x being Point of M holds x in Cl P iff (dist_min P) . x = 0
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, x be
  Point of M;
  hereby
    assume x in Cl P;
    then for a being Real st a > 0 ex p being Point of M st p in P &
    dist (x, p) < a by Th6;
    hence (dist_min P) . x = 0 by Th7;
  end;
  assume (dist_min P) . x = 0;
  then
  for a being Real st a > 0 ex p being Point of M st p in P & dist
  (x, p) < a by Th7;
  hence thesis by Th6;
end;
