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

theorem
  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 low_toler(f,A)[*] = low_toler(f,A)
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 p be object;
    assume p in low_toler(f,A)[*];
    then consider x,y being object such that
A2: x in X & y in X and
A3: p = [x,y] by ZFMISC_1:def 2;
    reconsider x9 = x, y9 = y as Element of X by A2;
    f.(x9,y9) <= A by A1,Th26;
    hence p in low_toler(f,A) by A3,Def3;
  end;
  then
  low_toler(f,A) c= low_toler(f,A)[*] & low_toler(f,A)[*] c= low_toler(f,A
  ) by LANG1:18;
  hence thesis by XBOOLE_0:def 10;
end;
