
theorem
  for M being non empty MetrSpace holds for x being Point of M holds for
  P being Subset of TopSpaceMetr(M) holds P <> {} & P is compact implies ex x2
being Point of TopSpaceMetr(M) st x2 in P & (dist(x)).x2 = lower_bound((dist(x)
  ).:P) by Th15,Th16;
