\subsection{Proof}

To verify that our approach of cumulative fusion operator ($\oplus$) of multinomial opinions is valid, we verify that the operator preserves the properties that valid multinomial opinions must respect.

\begin{proof}[Property 1: $b+d+u = 1$]
The point is to prove that the fusion equation for multinomial opinions preserve the properties : $b+d+u = 1$ and $\sum_{i=1}^3a_i^H=1$. This method leverages the existing cumulative fusion operator ($\oplus$) for binomial opinions to handle multinomial opinions.

Given two multinomial opinions in a ternary domain $W^F$ and $W^G$, with cardinality $k=3$:

\begin{align*}
W^F=&\left([b_1^F,b_2^F,b_3^F], u^H, [a_1^F,a_2^F,a_3^F] \right)\\
W^G=&\left([b_1^G,b_2^G,b_3^G], u^H, [a_1^G,a_2^G,a_3^G] \right)
\end{align*}


Each multinomial opinion can be represented into $k$ binomial opinions, where each binomial opinion focuses on one element of the ternary set versus the other two:

\begin{align*}
W^F&=
\begin{array}{rl}
W^F_1=&\left(b_1^F, d_1^F=(b_2^F+b_3^F), u^H, a_1^F \right)\\
W^F_2=&\left(b_2^F, d_2^F=(b_1^F+b_3^F), u^H, a_2^F \right)\\
W^F_3=&\left(b_3^F, d_3^F=(b_1^F+b_2^F), u^H, a_3^F \right)\\
\end{array}
\end{align*}

\begin{align*}
W^G&=
\begin{array}{rl}
W^G_1=&\left(b_1^G, d_1^G=(b_2^G+b_3^G), u^H, a_1^G \right)\\
W^G_2=&\left(b_2^G, d_2^G=(b_1^G+b_3^G), u^H, a_2^G \right)\\
W^G_3=&\left(b_3^G, d_3^G=(b_1^G+b_2^G), u^H, a_3^G \right)\\
\end{array}
\end{align*}

A valid multinomial opinions must satisfy the conditions outlined in Eq.~\eqref{eq:mn_bmd} and~\eqref{eq:mn_brd}. The underlying binomial opinions hence then also satisfy the condition in Eq.~\eqref{eq:add_req}. The resulting binomial opinions also adhere to their respective base rate $a$, which affects the uncertainty component.

We want to perform the cumulative fusion operator ($\oplus$) on the multinomial opinions to obtain $W^H = W^F \oplus W^G$, 

\begin{equation*}
    W^H = \left( [b_1^H,b_2^H,b_3^H], u^H, [a_1^H,a_2^H,a_3^H] \right)
\end{equation*}


To achieve this we perform cumulative fusion for each pair of corresponding binomial opinions from the two multinomial opinions $W^F$ and $W^G$. The mapping involves treating each option in isolation against the others.


\begin{align*}
W^H&=
\begin{array}{rl}
W^H_1=&\left(b_1^H, d_1^H, u^H, a_1^H \right) = W^F_1 \oplus  W^G_1\\
W^H_2=&\left(b_2^H, d_2^H, u^H, a_2^H \right) = W^F_2 \oplus  W^G_3\\
W^H_3=&\left(b_3^H, d_3^H, u^H, a_3^H \right) = W^F_3 \oplus  W^G_3\\
\end{array}
\end{align*}


The equations governing cumulative fusion of binomial opinions are listed in Eq.~\eqref{eq:cum_fus}. By applying the cumulative fusion to binomial opinions, we ensure that each pair of corresponding binomial opinions from two multinomial opinions is combined in a way that respects the subjective logic constraints.

The reconstructed multinomial opinion from the $k$ fused binomial opinions should also adhere to the constraints of subjective logic:

\begin{align*}
\sum_{i=1}^k (b_i + d_i + u_i) = k
\end{align*}


Let us consider the disbelief in the first component $d^H_1$.

\begin{align*}
d^H_1 &= \dfrac{d_1^Fu^G + d_1^Gu^F}{u^F+u^G-u^Fu^G} \\% because d_1 = b_2+b_3\\
      &= \dfrac{(b_2^F+b_3^F)^Fu^G + (b_2^G+b_3^G)u^F}{u^F+u^G-u^Fu^G}\\
      &= \dfrac{b_2^Fu^G+b_2^Gu^F}{u^F+u^G-u^Fu^G} + \dfrac{b_3^Fu^G+b_3^Gu^F}{u^F+u^G-u^Fu^G}\\
d^H_1 &= b_2^H + b_3^H
\end{align*}

By symmetry, we obtain the same results for the other components:
\begin{align*}
d^H_1 &= b_2^H + b_3^H\\
d^H_2 &= b_1^H + b_3^H\\
d^H_3 &= b_1^H + b_2^H
\end{align*}


From these expressions of the disbeliefs of $W^H$ using the beliefs of $W^F$ and $W^G$, we can evaluate the following goal equation $b^H+d^H+u^H=1$ for the first component:

\begin{align*}
&d_1^H = 1-u^H - b_1^H\\
&b_2^H + b_3^H = 1- \dfrac{u^Fu^G}{u^F+u^G-u^Fu^G} - \dfrac{b_1^Fu^G+b_1^Gu^F}{u^F+u^G-u^Fu^G}\\
%&b_2^Fu^G+b_2^Gu^F + b_3^Fu^G+b_3^Gu^F = u^F+u^G-u^Fu^G - u^fu^G - b_1^Fu^G-b_1^Gu^F\\
&\underbrace{(b_1^F+b_2^F+b_3^F-1)}_{\triangleq 1}u^G + \underbrace{(b_1^G+b_2^G+b_3^G-1)}_{\triangleq 1}u^F = -2u^Fu^G\\
&-u^Fu^G -u^Fu^G = -2u^Fu^G
\end{align*}

From here we can prove that

\begin{flalign*}
&\sum_{i=1}^k (b_i + d_i + u_i) = k \quad here \hspace{0.5em}  k=3 &\\
&b_1^H + b_2^H + b_3^H + d_1^H + d_2^H + d_3^H + u^H + u^H + u^H = 3 &\\
&b_1^H + b_2^H + b_3^H + b_2^H + b_3^H + b_1^H + b_3^H + b_1^H + b_2^H + 3 u^H = 3 &\\
&3 b_1^H + 3 b_2^H + 3 b_3^H + 3 u^H = 3 &\\
&3 (b_1^H + b_2^H + b_3^H + u^H) = 3 &\\
&3 (1) = 3 &\\
&3 = 3 &
\end{flalign*}

We now verified that the goal relationship of the ternary belief function is respected when using the multinomial fusion functions provided above.
\end{proof}

\begin{proof}[Property 2: $\sum_{i=1}^ka_i=1$]
For the second property, we start with the known relationship with the two ternary belief functions:

\begin{align*}
\sum_{i=1}^3a_i^F=&1\\
\sum_{i=1}^3a_i^G=&1
\end{align*}

Through the fusion, the $ai$ value of the new function is expressed by

\begin{equation*}
a_1^H = \dfrac{a_1^Fu^G+a_1^Gu^F - (a_1^F+a_1^G)u^Fu^G}{u^F+u^G-2u^Fu^G}
\end{equation*}

From this equation, we can express the sum of all $a$ values of the new belief:

\begin{equation*}
\sum_{i=1}^3a_i^F=\dfrac{u^G\overbrace{\sum\limits_{i=1}^3a_i^F}^{\triangleq 1} + u^F\overbrace{\sum\limits_{i=1}^3a_i^G}^{\triangleq 1} -u^Fu^G(\overbrace{\sum\limits_{i=1}^3(a_i^F+a_i^G}^{\triangleq 2}))}{u^F+u^G-2u^Fu^G}
\end{equation*}

We can verify that the property on the $a$ values is respected.
\end{proof}