Lower bound

Lower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower boundLower bound