reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th41:
  for M being bounded non empty MetrSpace holds
  {the carrier of M} in fam_class_metr(M)
proof
  let M be bounded non empty MetrSpace;
  set a = diameter [#]M;
  the distance of M is symmetric by METRIC_1:def 8;
  then low_toler(the distance of M,a) is_symmetric_in the carrier of M by Th17;
  then
A1: dist_toler(M,a) is_symmetric_in the carrier of M by Th33;
  the distance of M is Reflexive by METRIC_1:def 6;
  then low_toler(the distance of M,a) is_reflexive_in the carrier of M by Th16;
  then dist_toler(M,a) is_reflexive_in the carrier of M by Th33;
  then dom dist_toler(M,a) = the carrier of M by Th3;
  then the carrier of M c= field dist_toler(M,a) by XBOOLE_1:7;
  then reconsider R = dist_toler(M,a)[*] as Equivalence_Relation of M
  by A1,Th9;
  Class R = {the carrier of M} by Th40;
  hence thesis by Def8;
end;
