For example, the language formula_34 can be shown to be non-context-free by using the pumping lemma in a proof by contradiction. First, assume that is context free. By the pumping lemma, there exists an integer which is the pumping length of language . Consider the string formula_35 in . The pumping lemma tells us that can be written in the form formula_7, where , and are substrings, such that formula_10, formula_11, and formula_39 for every integer formula_40. By the choice of and the fact that formula_11, it is easily seen that the substring can contain no more than two distinct symbols. That is, we have one of five possibilities for :