Let $\displaystyle G=\langle t, A|t^{-1}Ht=K \rangle$ be an HNN extension with $\displaystyle H, K \in Z(A)$.

Let $\displaystyle x,y \in A$ such that $\displaystyle x \notin H \cup K$.

Then $\displaystyle x \nsim _{G} y$ if and only if $\displaystyle x \nsim_{A} y$.

It is obvious that if $\displaystyle x \nsim_{G} y$ then $\displaystyle x \nsim_{A} y$ since $\displaystyle A \subset G$.

I tried to prove the opposite direction but I have no idea where to use $\displaystyle H, K \in Z(A)$ in the proof.

Any hint??

Thank you.