
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 x1
being Point of TopSpaceMetr(M) st x1 in P & (dist(x)).x1 = upper_bound((dist(x)
  ).:P) by Th14,Th16;
