
theorem
  for M being non empty MetrSpace, P, Q being non empty Subset of
  TopSpaceMetr M st P is compact & Q is compact holds min_dist_max (P, Q) >= 0
proof
  let M be non empty MetrSpace, P, Q be non empty Subset of TopSpaceMetr M;
  assume P is compact & Q is compact;
  then ex x1, x2 being Point of M st x1 in P & x2 in Q & dist( x1,x2) =
  min_dist_max (P,Q) by WEIERSTR:31;
  hence thesis by METRIC_1:5;
end;
