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

theorem Th35:
  for M being Reflexive symmetric non empty MetrStruct holds
  fam_class_metr(M) = fam_class(the distance of M)
proof
  let M be Reflexive symmetric non empty MetrStruct;
  now
    let z be object;
    assume z in fam_class(the distance of M);
    then consider
    a being non negative Real, R be Equivalence_Relation of
    the carrier of M such that
A1: R = low_toler(the distance of M,a)[*] and
A2: Class(R) = z by Def5;
    reconsider R1 = R as Equivalence_Relation of M;
    R1 = dist_toler(M,a)[*] by A1,Th33;
    hence z in fam_class_metr(M) by A2,Def8;
  end;
  then
A3: fam_class(the distance of M) c= fam_class_metr(M);
  now
    let z be object;
    assume z in fam_class_metr(M);
    then consider
    a being non negative Real, R be Equivalence_Relation of
    M such that
A4: R = dist_toler(M,a)[*] and
A5: Class(R) = z by Def8;
    R = low_toler(the distance of M,a)[*] by A4,Th33;
    hence z in fam_class(the distance of M) by A5,Def5;
  end;
  then fam_class_metr(M) c= fam_class(the distance of M);
  hence thesis by A3,XBOOLE_0:def 10;
end;
