below) of the regions obtained by intersecting opposite wedge–shaped regions in the ambient space $\RR^{n+1}$ with , we call them opposite wedges. In general, if $W = \lambda W_R$, for some $\lambda \in {\mbox{SO$_0(2,n\!-\!1)$}}$, then $W' = \lambda W_R{}'$.

In [@AdS] it was shown that for $n \geq 3$ the above results entail that observables which are localized in complementary wedges are weakly local in the vacuum state. So we can state:

\[weaklocality1\] Under the assumptions (i)–(iv) and for $n \geq 3$, observables which are localized in opposite wedges of are weakly local with respect to each other. Explicitly, for any $\lambda \in {\mbox{SO$_0(2,n\!-\!1)$}}$ and any $A' \in \As(\lambda W_R{}')$, $B \in \As(\lambda W_R)$, one has $\langle\Omega,A'B\Omega\rangle = \langle\Omega,BA'\Omega\rangle$.

This result is also valid for theories in the covering space of which satisfy (i)–(iv). A similar result was proven in [@BEM] for two–point functions of quantum fields on the covering space of satisfying a different set of assumptions in the Wightman framework.

The uniqueness of the modular objects and the covariance assumption (iii) give us a relation we shall use repeatedly in the following: $$\label{unique}
U(\lambda) J_W U(\lambda)^{-1} = J_{\lambda W} \, ,$$ for all $W \in \Ws$ and all 