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

theorem
  for M being bounded non empty MetrSpace holds
  fam_class_metr(M) is Strong_Classification of the carrier of M
proof
  reconsider a = 0 as non negative Real;
  let M be bounded non empty MetrSpace;
  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 in fam_class_metr(M) by Def8;
  then
A2: SmallestPartition (the carrier of M) in fam_class_metr(M) by Th36;
  fam_class_metr(M) is Classification of the carrier of M & {the carrier
  of M} in fam_class_metr(M) by Th41,Th42;
  hence thesis by A2,Def2;
end;
