
theorem
  for M being non empty MetrSpace, P being non empty Subset of
TopSpaceMetr M, z being Point of M st P is compact holds (dist_min P) . z <= (
  dist_max P) . z
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, z be
  Point of M;
  consider w being Point of M such that
A1: w in P and
A2: (dist_min P) . z <= dist (w, z) by Th19;
  assume P is compact;
  then (dist_max P) . z >= dist (z, w) by A1,Th20;
  hence thesis by A2,XXREAL_0:2;
end;
