For comparison with the pumping lemma proof that the "a^n b^n" language is not regular, I give a proof using the Myhill theorem.

To prove the "a^n b^n" language is not a regular language using this theorem, we must demonstrate an infinite number of prefixes with incompatible suffix sets S(x). But these are trivially available: if we take x_n=a^n, then S(x_n) contains the word b^n; this suffix is not in S(x_m) for any other m. So we have an infinite number of types for S(x), thus the language is not regular.