I am trying to prove by induction that if
$$\hat\delta_d(q_0, w) = \hat\delta_n(q_0, w)$$
I know by practicing inductive proofs of the $\hat\delta$ for DFAs, that on the basis of the definition of $\hat\delta_d$
for any $q \in Q$, $\hat\delta(q, \epsilon) = q$
for any $q \in Q$ and $a \in $ the language of the DFA, $\hat\delta(\delta(q, y), a)$
$\hat\delta(q, ax) = \hat\delta(q_1, x)$ where $a_1 = \delta(q,a)$
Does the extended transition function also have these properties? Does it have other definitions/properties?
For reference, $\hat\delta_d$ refers to the extended transition function of DFAs and $\hat\delta_n$ refers to the extended transition function of NFAs.