
theorem Th22:
  for M being non empty MetrSpace, P, Q being non empty Subset of
  TopSpaceMetr M, z being Point of M st P is compact & Q is compact & z in Q
  holds (dist_min P) . z <= max_dist_max (P, Q)
proof
  let M be non empty MetrSpace, P, Q 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 & Q is compact & z in Q;
  then dist (w, z) <= max_dist_max (P, Q) by A1,WEIERSTR:34;
  hence thesis by A2,XXREAL_0:2;
end;
