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

theorem Th24:
  for X being non empty set, f being PartFunc of [:X,X:],REAL,
      R being Equivalence_Relation of X st R = low_toler(f,0)[*] &
  f is nonnegative Reflexive discerning holds R = id X
proof
  let X be non empty set, f be PartFunc of [:X,X:],REAL, R be
  Equivalence_Relation of X such that
A1: R = low_toler(f,0)[*] and
A2: f is nonnegative Reflexive discerning;
A3: for x,y being object st x in X & x = y holds [x,y] in low_toler(f,0)[*]
  proof
    let x,y be object;
    assume x in X & x = y;
    then [x,y] in low_toler(f,0) by A2,Th21;
    hence thesis by A2,Th23;
  end;
  for x,y being object st [x,y] in low_toler(f,0)[*] holds x in X & x = y
  proof
    let x,y be object;
    assume [x,y] in low_toler(f,0)[*];
    then [x,y] in low_toler(f,0) by A2,Th23;
    hence thesis by A2,Th20,ZFMISC_1:87;
  end;
  hence thesis by A1,A3,RELAT_1:def 10;
end;
