Michael Sipser Edition 3 Exercise 1 Question 57 (Page No. 92)

Question

If $A$ is any language,let $A_{\frac{1}{2}-}$ be the set of all first halves of strings  in $A$ so that $A_{\frac{1}{2}-}=\{\text{x|for some y,|x|=|y| and xy $\in$ A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}-}.$