
theorem Th23:
  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_max 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;
  assume that
A1: P is compact and
A2: Q is compact;
  reconsider P as non empty compact Subset of TopSpaceMetr M by A1;
  set A = (dist_max P) .: Q;
A3: dom dist_max P = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  assume z in Q;
  then
A4: (dist_max P) . z in A by A3,FUNCT_1:def 6;
  upper_bound ((dist_max P) .: Q) = max_dist_max (P, Q) by WEIERSTR:def 10;
  then consider X being non empty Subset of REAL such that
A5: A = X and
A6: max_dist_max (P, Q) = upper_bound X by Th11;
  [#]A is real-bounded by A2,WEIERSTR:9,11;
  then X is real-bounded by A5,WEIERSTR:def 1;
  then X is bounded_above;
  hence thesis by A5,A6,A4,SEQ_4:def 1;
end;
