3. Let $M = (Q, \Sigma, \delta, s, F)$ be an arbitrary DFA. Prove by induction on $|y|$ that for all strings $x, y \in \Sigma^*$ and $q \in Q$,
$\hat{\delta}(q, xy) = \hat{\delta}(\hat{\delta}(q, x), y)$,
where $\hat{\delta}$ is the extended version of $\delta$ defined on all strings