
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 x2 being Point of TopSpaceMetr(M) st x2 in Q & (dist_max(P)).x2 =
  lower_bound((dist_max(P)).:Q) by Th15,Th24;
