
theorem
  for M being non empty MetrSpace holds for P,Q being Subset of
  TopSpaceMetr(M) holds P <> {} & P is compact & Q <> {} & Q is compact implies
  ex x1 being Point of TopSpaceMetr(M) st x1 in Q & (dist_min(P)).x1 =
  upper_bound((dist_min(P)).:Q) by Th14,Th27;
