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

theorem
  for X being non empty set, f being PartFunc of [:X,X:],REAL, a being
non negative Real st low_toler(f,a) is_reflexive_in X & f is symmetric
  holds fam_class(f) is non empty
proof
  let X be non empty set, f be PartFunc of [:X,X:],REAL, a be non negative
  Real;
  assume low_toler(f,a) is_reflexive_in X & f is symmetric;
  then reconsider R = low_toler(f,a)[*] as Equivalence_Relation of X by Th22;
  reconsider x = Class(R) as set;
  x in fam_class(f) by Def5;
  hence thesis;
end;
