the parenthesis.[^5]

(i)  $t > 48\pi^2{{(N-1)^3}\over{N(N^2-3)}}$

We immediately observe that the scale $\mbar_2$ $(\mbar_3)$ is the phase boundary between the Hosotani phase and the coexisting one (the coexisting phase and the Higgs one). There is no overlapping region of the phases for this parameter region of $t$. Thus, the vacuum configuration is given by $$\mbox {vacuum configuration}=\left\{\begin{array}{lll}
\mbox{Hosotani phase} &\mbox{for}& \mbar < \mbar_2,\\
\mbox{coexisting phase} &\mbox{for}& \mbar_2 < \mbar < \mbar_3,\\
\mbox{Higgs phase} & \mbox{for}& \mbar_3 < \mbar.
\end{array}\right.
\label{solu}$$ The order parameters in the Hosotani and coexisting phases (the coexisting and Higgs phases) are connected continuously at the phase boundary $\mbar_2$ $(\mbar_3)$, so that the phase transition is the second order.

(ii)  $24\pi^2{{(N-1)^3}\over {N^2(2N-3)}}
< t \leq  48\pi^2{{(N-1)^3}\over{N(N^2-3)}}$

For this parameter region of $t$, the Hosotani and coexisting phases overlap between $\mbar_1$ and $\mbar_2$. Let us consider the quantity $\Delta\Vb\equiv\Vb_{Hosotani}-\Vb_{coexisting}$, which is a monotonically increasing function with respect to $\bar{m}$, $${{\del\Delta\Vb}\over{\del\mbar^2}}=\half \vbar^2(\mbar)\geq 0,
\label{monou}$$ as shown in the appendix. By denoting the scale giving $\Vb_{coexisting}=\Vb_{Hosotani}$ by $\mbar_4$, we can conclude that for $\mbar \leq \mbar_4$ $(\mbar_4 < \mbar\leq \mbar_3)$, the Hosotani (coexisting) phase is realized as the vacuum configuration. The Higgs phase can exist for $ \mbar_3 < \mbar$. 