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

theorem Th22:
  for X being non empty set, f being PartFunc of [:X,X:],REAL,
      a being Real st low_toler(f,a) is_reflexive_in X & f is symmetric holds
   low_toler(f,a)[*] is Equivalence_Relation of X
proof
  let X be non empty set, f be PartFunc of [:X,X:],REAL, a be Real such
  that
A1: low_toler(f,a) is_reflexive_in X and
A2: f is symmetric;
    dom low_toler(f,a) = X by A1,Th3;
    then
AA: X c= field low_toler(f,a) by XBOOLE_1:7;
  now
    let x,y be object such that
A3: x in X & y in X and
A4: [x,y] in low_toler(f,a);
    reconsider x1 = x, y1 = y as Element of X by A3;
    f.(x1,y1) <= a by A4,Def3;
    then f.(y1,x1) <= a by A2,METRIC_1:def 4;
    hence [y,x] in low_toler(f,a) by Def3;
  end;
  then low_toler(f,a) is_symmetric_in X by RELAT_2:def 3;
  hence thesis by AA,Th9;
end;
