
theorem Th31:
  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 <= max_dist_min (R, Q)
proof
  let M be non empty MetrSpace, Q, R be non empty Subset of TopSpaceMetr M, y
  be Point of M;
  assume that
A1: Q is compact & R is compact and
A2: y in Q;
  set A = (dist_min R) .: Q;
  consider X being non empty Subset of REAL such that
A3: A = X and
A4: upper_bound A = upper_bound X by Th11;
  dom dist_min R = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then
A5: (dist_min R).y in X by A2,A3,FUNCT_1:def 6;
  max_dist_min (R, Q) = upper_bound ((dist_min R) .: Q) & X is
  bounded_above by A1,A3,Th25,WEIERSTR:def 8;
  hence thesis by A4,A5,SEQ_4:def 1;
end;
