\leq\|(V_{\Phi}-V_{\Psi})\otimes I_R \cdot\rho\cdot V^*_{\Phi}\otimes I_R\|_1+\|V_{\Psi}\otimes I_R \cdot\rho\cdot(V^*_{\Phi}-V^*_{\Psi})\otimes I_R\|_1\\\\
\leq\|(V_{\Phi}-V_{\Psi})\otimes I_R\|_E^{G\otimes I_R} \|V_{\Phi}\otimes I_R\|_E^{G\otimes I_R}+\|(V_{\Phi}-V_{\Psi})\otimes I_R\|_E^{G\otimes I_R}\|V_{\Psi}\otimes I_R\|_E^{G\otimes I_R}
\\\\
\leq\|V_{\Phi}-V_{\Psi}\|_E^G \|V_{\Phi}\|_E^G+\|V_{\Phi}-V_{\Psi}\|_E^G \|V_{\Psi}\|_E^G.
\end{array}$$ The first and the second inequalities follow from the properties of the trace norm (the non-increasing under partial trace and the triangle inequality), the third inequality follows from Proposition \[bp-en-0\]D, the last one – from Proposition \[pen-tp\]A. By noting that $[\|V_{\Phi}\|^G_E]^2=\|\Phi\|^G_{\mathrm{cb},E}$ and $[\|V_{\Psi}\|^G_E]^2=\|\Psi\|^G_{\mathrm{cb},E}$ we obtain the first inequality in (\[KSW-rel\]).

To prove the second inequality in (\[KSW-rel\]) note that $\beta_E^G(\Phi,\Psi)\leq\sqrt{\|\Phi-\Psi\|^G_{\mathrm{cb},E}}$. This follows from the inequality $\beta(\rho,\sigma)\leq\sqrt{\|\rho-\sigma\|_1}$ valid for any $\rho$ and $\sigma$ in $\T_+(\H)$, which is easily proved by using the inequality ${\mathrm{Tr}}(\sqrt{\rho}-\sqrt{\sigma})^2\leq \|\rho-\sigma\|_1$ (see the proof of Lemma 9.2.3 in [@H-SCI]). So, it suffices to show that $$\label{r-r}
\inf_{V_{\Phi},V_{\Psi}}\|V_{\Phi}-V_{\Psi}\|_E^G=\beta_E^G(\Phi,\Psi).$$

Denote by $\alpha^G_E(\Phi,\Psi)$ the l.h.s. of (\[r-r\]). Let $\C^s_{{G},E}$ be the subset of $\S(\H_A)$ determined by the inequality $\,{\mathrm{Tr}}{G}\rho\leq E\,$ and $\N(\Phi,\Psi)=\bigcup V_{\Phi}^*V_{\Psi}$, where the union is over *all* common Stinespring representations (\[c-S-r\]). Then by using definition (\[ec-on\]) we obtain $$\label{beta-e}
\alpha^G_E(\Phi,\Psi)=\inf_{N\in\N(\Phi,\Psi)} \sup_{\rho\in\C^s_{{G},E}}\sqrt{{\mathrm{Tr}}{\hspace{1pt}}\Phi(\rho)+{\mathrm{Tr}}\Psi(\rho)-2\Re{\hspace{1pt}}{\mathrm{Tr}}N\rho}.$$ Following the proof of Theorem 1 in show that $\N(\Phi,\Psi)$ coincides with the set $$\M(\Phi,\Psi)\doteq \left\{V_{\Phi} ^*(I_B\otimes C)V_{\Psi}\,|\,C\in \B(\H_E), \|C\|\leq 1\right\},$$ defined via some *fixed* common Stinespring representation (\[c-S-r\]). It will imply, in particular, 