$\backslash$Cut the Edges.

$($a$)$ If $Z$ is any language, define the language ECut$(Z)$ to be where $a,$bin

$\{$:$,$win${}^{**}\}.$ Prove that if $Z$ is regular, then so is ECut$(Z).$

$($b$)$ Similarly, define the language UCut$(Z)$ to be $\{\begin{array}{ccc}w\text{:}\text{A}\text{A}\text{a}\text{:}& \text{A}\text{A}\text{b}\text{:}& \text{a}\text{w}\text{b}\text{i}\text{n}\text{Z}\end{array}$ where $\{$:$a,$bin$,$win${}^{**}\}.$

Prove that if $Z$ is regular, then so is UCut$(Z).$