Prove $\{a^ib^i\mid i\ge0\}$ is not regular using the pumping lemma

Question

I do not understand the last sentence of the proof provided. It says that the fact that xz does not belong to L contradicts the hypothesis, but isn't it that xyz not belonging to L what we are trying to prove?

Answer 1

By the pumping lemma $a^n b^n$ can be written as $xyz$ with $|xy| \le n$ and $|y| \ge 1$ in a way that ensures that $xy^kz \in L$ for any $k \ge 0$. Picking $k=0$, this implies that $xz \in L$.

Since this is not the case for your language $L$, you know that $L$ cannot be regular.