
theorem Th29:
  for M being non empty MetrSpace, P being non empty Subset of
  TopSpaceMetr M holds max_dist_min (P, P) = 0
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M;
A1: [#] ((dist_min P).:P) = (dist_min P) .: P by WEIERSTR:def 1
    .= { 0 } by Th27;
  max_dist_min (P, P) = upper_bound ((dist_min P).:P) by WEIERSTR:def 8
    .= upper_bound { 0 } by A1,WEIERSTR:def 2;
  hence thesis by SEQ_4:9;
end;
