Define a function $\delta_{1}:S\times\Sigma\to P(S)$, as follows: for each pair $(s,a)\in S\times\Sigma$, let

In other words, $\delta_{1}(s,a)$ is the set of all states reachable from $s$ by words of the form $\epsilon^{m}a\epsilon^{n}$. As usual, we extend $\delta_{1}$ so its domain is $S\times\Sigma^{*}$. By abuse of notation, we use $\delta_{1}$ again for this extension. First, we set $\delta_{1}(s,\epsilon):=\{s\}$. Then we inductively define $\delta_{1}(s,ua)=\delta_{1}(\delta_{1}(s,u),a)$. Using induction,

So for any non-empty word $u$, we have the following equation:

In other words, if $u=a_{1}a_{2}\cdots a_{n}$, then $\delta_{1}(s,u)$ is the set of all states reachable from $s$ by words of the form

Now, define $A$ to be the automaton $(S,\Sigma,\delta_{1},I,F)$. Then, from equation (1) above, a word

is accepted by $A$ iff some word $v$ of the form (2) is accepted by $E_{{\epsilon}}$ iff $u=h(v)$ is accepted by $E$, proving the proposition. ∎