We defined & by breaking the input string into any string followed by a single symbol $($in the inductive part, Equation $2\mathrm{.}1).$ However, we informally think of & as describing what happens along a path with a certain string of labels, and if so$,$ then it should not matter how we break the input string in the definition of &$.$ Show that in fact, $($q$,$zy$)=8(8($q$,7),$y$)$ for any state q and strings x and y$.$ Hint: Perform an induction on $|$y$|.$