theorem Th36:
  for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is
  compact & Q <> {} & Q is compact holds min_dist_min(P,Q)>=0
proof
  let P,Q be Subset of TopSpaceMetr(M);
  assume P <> {} & P is compact & Q <> {} & Q is compact;
  then ex x1,x2 being Point of M st x1 in P & x2 in Q & dist( x1,x2) =
  min_dist_min(P,Q) by WEIERSTR:30;
  hence thesis by METRIC_1:5;
end;
