Let G=\langle t, A|t^{-1}Ht=K \rangle be an HNN extension with H, K \in Z(A).
Let x,y \in A such that x \notin H \cup K.
Then x \nsim _{G} y if and only if x \nsim_{A} y.

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

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

Any hint??

Thank you.