$$\label{beta-d}
\begin{array}{rl}
\alpha^G_E(\Phi,\Psi)&
=\displaystyle\inf_{C\in\B_1(\H_E)}\sup_{\rho\in\C^s_{{G},E}}\sqrt{{\mathrm{Tr}}{\hspace{1pt}}\Phi(\rho)+{\mathrm{Tr}}\Psi(\rho)-2\Re{\hspace{1pt}}{\mathrm{Tr}}V_{\Phi}^*(I_B\otimes C)V_{\Psi}\rho}\\
&=\displaystyle\sup_{\rho\in\C^s_{{G},E}}\inf_{C\in\B_1(\H_E)} \sqrt{{\mathrm{Tr}}{\hspace{1pt}}\Phi(\rho)+{\mathrm{Tr}}\Psi(\rho)-2\Re{\hspace{1pt}}{\mathrm{Tr}}V_{\Phi} ^*(I_B\otimes C)V_{\Psi}\rho}\\
&=\displaystyle\sup_{\rho\in\C^s_{{G},E}}\sqrt{{\hspace{1pt}}{\mathrm{Tr}}{\hspace{1pt}}\Phi(\rho)+{\mathrm{Tr}}\Psi(\rho)-2\sup_{C\in\B_1(\H_E)}|{\mathrm{Tr}}V_{\Phi} ^*(I_B\otimes C)V_{\Psi}\rho|},
\end{array}$$ where the possibility to change the order of the optimization follows from Ky Fan’s minimax theorem [@Simons] and the $\sigma$-weak compactness of the unit ball $\B_1(\H_E)$ of $\B(\H_E)$ . It is easy to see that $$\label{a-eq}
\sup_{C\in\B_1(\H_E)}|{\mathrm{Tr}}V_{\Phi} ^*(I_B\otimes C)V_{\Psi}\rho|=\!\!\sup_{C\in\B_1(\H_E)}|\langle V_{\Phi}\otimes I_R{\hspace{1pt}}\varphi|I_{BR}\otimes C |V_{\Psi}\otimes I_R {\hspace{1pt}}\varphi\rangle|,\!$$ where $\varphi$ is a purification of $\rho$, i.e. a vector in $\H_{A}\otimes\H_R$ such that ${\mathrm{Tr}}_R|\varphi\rangle\langle\varphi|=\rho$.

Since for *any* common Stinespring representation (\[c-S-r\]) and any purification $\varphi$ of a state $\rho$ the vectors $V_{\Phi}\otimes I_R{\hspace{1pt}}|\varphi\rangle$ and $V_{\Psi}\otimes I_R{\hspace{1pt}}|\varphi\rangle$ in $\H_{BER}$ are purifications of the operators $\Phi\otimes {\mathrm{Id}}_R(|\varphi\rangle\langle\varphi|)$ and $\Psi\otimes {\mathrm{Id}}_R(|\varphi\rangle\langle\varphi|)$ in $\T(\H_{BR})$, by using the relation $\N(\Phi,\Psi)=\M(\Phi,\Psi)$ proved before and Uhlmann’s theorem [@Uhlmann; @Wilde] it is easy to show that the square of the r.h.s. of (\[a-eq\]) coincides with the fidelity of these operators defined in (\[fidelity\]). Note also that ${\mathrm{Tr}}{\hspace{1pt}}\Phi\otimes {\mathrm{Id}}_R(\sigma)={\mathrm{Tr}}{\hspace{1pt}}\Phi(\sigma_A)$ and ${\mathrm{Tr}}\Psi\otimes {\mathrm{Id}}_R(\sigma)={\mathrm{Tr}}\Psi(\sigma_A)$ for any state $\sigma$ in $\S(\H_{AR})$. By Remark \[ec-b-dist-r\] these observations and (\[beta-d\]) imply that $\alpha^G_E(\Phi,\Psi)=\beta_E^G(\Phi,\Psi)$, i.e. that (\[r-r\]) holds.

The last assertion can be derived from the attainability of the infimum in the first line in (\[beta-d\]) which follows from the $\sigma$-weak compactness of the unit ball 