$15\mathrm{.}1$ Analyzing Two$-$Way DFA$$s

Let M be a two$-$way DFA, let p be a state of M$,$ and let u and v be two arbitrary strings such that M leaves the input to the right in state p when started on either u or v$.$ Then u and v must be L$($M$)-$equivalent in the sense of the Myhill$-$Nerode Theorem.