\mbox{for} \ \ A' \in {{\cal A}}(\widetilde{W}_R{}') \, .$$ From equations (\[unique\]) and (\[WR0n\]) follow the equalities $$\label{0nR}
J_{\widetilde{W}_R{}'} = e^{i\pi M_{0n}} J_{W_R} e^{-i\pi M_{0n}} = 
e^{i2\pi M_{0n}}J_{W_R} = J_{W_R} 
\, ,$$ where we have used $e^{i2\pi M_{0n}} = 1$, valid in proper AdS but not in its covering space. Thus, by (\[equation15\]) we have for $n \geq 3$: $$\label{Js}
J_{W_R{}'} = J_{W_R} = J_{\widetilde{W}_R} = J_{\widetilde{W}_R{}'} \, .$$ We can now prove that $\As(\widetilde{W}_R{}')$ and $\As(W_R)$ are weakly local with respect to each other.

\[weaklocality2\] Under the assumptions (i)–(iv) and for $n \geq 3$, observables which are localized in $W_R$ are weakly local with respect to observables localized in $\widetilde{W}_R{}'$. Explicitly, for any $\lambda \in {\mbox{SO$_0(2,n\!-\!1)$}}$ and any $A' \in \As(\lambda \widetilde{W}_R{}')$, $B \in \As(\lambda W_R )$, one has $\langle\Omega,A'B\Omega\rangle = \langle\Omega,BA'\Omega\rangle$.

With the above preparations, one sees that for any $A' \in \As(\widetilde{W}_R{}')$ and $B \in \As(W_R )$, one has $$\begin{split}
{{\langle\Omega }}, A^{\prime *} B {{\Omega \rangle}}& =  \overline{{{\langle\Omega }}, B^{*} A^{\prime}{{\Omega \rangle}}}
= \overline{{{\langle\Omega }}, B^{*} J_{W_R} J_{W_R} A^{\prime}{{\Omega \rangle}}}  \\ & = 
{{\langle\Omega }}, B e^{\, -\pi M_{0 1}} e^{\, \pi M_{0 1}} A^{\prime *} {{\Omega \rangle}}=  {{\langle\Omega }}, B A^{\prime *} {{\Omega \rangle}}\, .
