**Theorem:**Let be a regular language, and be a string. Then there exists a constant s.t. . We can break into three strings, , s.t.:

**Method to prove that a language is not regular:**

- At first, we have to assume that is regular.
- So, the pumping lemma should hold for .
- Use the pumping lemma to obtain a contradiction:
- Select s.t. .
- Select s.t. .
- Select s.t.
- Assign the remaining string to .
- Select s.t. the resulting string is not in .

**Problem:**Prove that is not regular.

**Solution:**

- At first, we assume that is regular and is the number of states.
- Let . Thus .
- By the pumping lemma, let , where .
- Let , , and , where , , , . Thusly .
- Let . Then .
- Number of .
- Hence, . Since , is not of the form !!!!!!!!!!!!!!!
- Thus, . Hence is not regular.