given：
Ω={θ_1,θ_2,⋯,θ_N} 2^Ω={∅,{θ_1 },⋯,{θ_N },{θ_1,θ_2 },⋯,{θ_1,θ_2,⋯,θ_n },⋯,Ω}

m_i: 2^Ω→[0,1]

\begin{aligned}
& \operatorname{BetP}_m\left(\theta_n\right)=\sum_{\theta_n \in A, A \subseteq \Omega} \frac{1}{|A|} \frac{m(A)}{1-m(\varnothing)} \\
& d_{BPA_{i j}}=d_{BPA}\left(m_i, m_j\right)=\sqrt{\frac{1}{2}\left(\left\|\vec{m}_i\right\|^2+\left\|\vec{m}_j\right\|^2-2\left\langle\vec{m}_i, \vec{m}_j\right\rangle\right)} \\
& \left\langle\vec{m}_i, \vec{m}_j\right\rangle=\sum_{A \subseteq 2^{\Omega}} \sum_{B \subseteq 2^{\Omega}} m_i(A) \cdot m_j(B) \cdot \frac{|A \cap B|}{|A \cup B|} \\
& k_{i j}^{\prime}=\sum_{\theta_n \cap \theta_{n^{\prime}}=\emptyset} \operatorname{BetP}_{m_i}\left(\theta_n\right) \cdot \operatorname{BetP}_{m_j}\left(\theta_{n^{\prime}}\right) \\
& \operatorname{Diss}_{BetP_{m_{i j}}}=\frac{1}{2} \sum_{n=1}^N\left|\operatorname{BetP}_{m_i}\left(\theta_n\right)-\operatorname{BetP}_{m_j}\left(\theta_n\right)\right| \\
& \operatorname{Diss}\left(m_i, m_j\right)=\alpha_{k_{i j}^{\prime}} \cdot k_{i j}^{\prime}+\alpha_{d_{BPA_{i j}}} \cdot d_{BPA_{i j}} \\
& \alpha_{k_{i j}^{\prime}}=\operatorname{Diss}_{BetP_{m_{i j}}}, \alpha_{d_{BPA_{i j}}}=\operatorname{Sim}_{BetP_{m_{i j}}}=1-\alpha_{k_{i j}^{\prime}} \\
&
\end{aligned}

please prove the following inequality ( display the process in latex fomat):
\text { (d6) Triangle inequality: } \operatorname{Diss}\left(m_i, m_j\right) \leq \operatorname{Diss}\left(m_i, m_g\right)+\operatorname{Diss}\left(m_g, m_j\right) \text {; }