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

theorem Th27:
  for X being finite non empty Subset of REAL, f being Function of
[:X,X:],REAL, z being finite non empty Subset of REAL, A being Real st z
  = rng f & A >= max z holds for R being Equivalence_Relation of X st R =
  low_toler(f,A)[*] holds Class R = {X}
proof
  let X be finite non empty Subset of REAL, f be Function of [:X,X:],REAL, z
  be finite non empty Subset of REAL, A be Real such that
A1: z = rng f & A >= max z;
  now
    let R be Equivalence_Relation of X such that
A2: R = low_toler(f,A)[*];
A3: for x being set st x in X holds X = Class(R,x)
    proof
      let x be set;
      assume x in X;
      then reconsider x9 = x as Element of X;
      now
        let x1 be object;
        assume x1 in X;
        then reconsider x19 = x1 as Element of X;
        f.(x19,x9) <= A by A1,Th26;
        then
A4:     [x1,x] in low_toler(f,A) by Def3;
        low_toler(f,A) c= low_toler(f,A)[*] by LANG1:18;
        hence x1 in Class(R,x) by A2,A4,EQREL_1:19;
      end;
      then X c= Class(R,x);
      hence thesis by XBOOLE_0:def 10;
    end;
    now
      let a be object;
      assume a in {X};
      then
A5:   a = X by TARSKI:def 1;
      consider x be object such that
A6:   x in X by XBOOLE_0:def 1;
      X = Class(R,x) by A3,A6;
      hence a in Class R by A5,A6,EQREL_1:def 3;
    end;
    then
A7: {X} c= Class R;
    now
      let a be object;
      assume a in Class R;
      then ex x being object st x in X & a = Class(R,x) by EQREL_1:def 3;
      then a = X by A3;
      hence a in {X} by TARSKI:def 1;
    end;
    then Class R c= {X};
    hence Class R = {X} by A7,XBOOLE_0:def 10;
  end;
  hence thesis;
end;
