
theorem Th32:
  for M being non empty MetrSpace, Q, R being non empty Subset of
  TopSpaceMetr M, y being Point of M st Q is compact & R is compact & y in Q
  holds (dist_min R).y <= HausDist (Q, R)
proof
  let M be non empty MetrSpace, Q, R be non empty Subset of TopSpaceMetr M, y
  be Point of M;
  assume Q is compact & R is compact & y in Q;
  then
  max_dist_min (R, Q) <= max (max_dist_min (R, Q), max_dist_min (Q, R)) &
  ( dist_min R).y <= max_dist_min (R, Q) by Th31,XXREAL_0:25;
  hence thesis by XXREAL_0:2;
end;
