
theorem Th9:
  for M being non empty MetrSpace, P being non empty closed Subset
  of TopSpaceMetr M, x being Point of M holds x in P iff (dist_min P) . x = 0
proof
  let M be non empty MetrSpace, P be non empty closed Subset of TopSpaceMetr M
  , x be Point of M;
  P = Cl P by PRE_TOPC:22;
  hence thesis by Th8;
end;
