 (4) 
where 
$$\sum_{X\in S^{\theta}}W_{m}(X)=1\,\,\,\mathrm{and~all}\,\,W_{m}(\cdot)\in[0,1],\,\mathrm{as~in~(3)}$$
and 

$$\sum_{z\in S_{\bigcap,r}^{\mathrm{even}}}V_{m}(z)=1\,\,\,\mathrm{and~all}\,\,\,V_{m}(\cdot)\in\left[0,1\right].$$
In the free and hybrid modes, if no non-empty intersection is redistributed, i.e. $S^{\rm non\Phi}_{\cap\nu}$
 and all Vm ( ) 0,1 ⋅ ∈[ ]. (5) 
contains no elements, DWO coincides with WO . In the Shafer's model, always DWO coincides with WO . 

 For s ≥ 2 sources, we have a similar formula: 

$\forall\ A\in\ \left(S^{\phi}\setminus S^{\text{mod}}_{\Gamma,r}\right)\setminus\{\Phi\},\ m_{\text{DTW}}(A)=\sum_{\begin{subarray}{c}X_{i},X_{i}\sim X_{i},\text{tr}\in\Gamma\\ \Gamma_{i}\sim A\end{subarray}}\prod_{i=1}^{r}m_{i}(X_{i})+W_{m}(A)\cdot m_{\text{tr}_{i}}(\Phi)+V_{m}(A)\cdot\sum_{z\in S^{\text{mod}}_{\Gamma,r}}m_{\text{tr}_{i}}(z)$

$$({\boldsymbol{\delta}})$$
$$\mathrm{\boldmath~\Psi~}\mathrm{\boldmath~\Psi~}W O\ .$$
 (6) 
with the same restrictions on ( ) Wm ⋅ and ( ) Vm ⋅ . 

## 3. A Fusion Rule Class Of Proportional Redistribution Of Intersection Masses

 For \ \, ( ) , { } non A SS I r t θ Φ ∈ Φ I for two sources we have: 

$$\begin{array}{r l}{{}}&{{}={\left(S^{\theta}\setminus S_{\cap,r}^{n o n\Phi}\right)}\backslash\left\{\Phi,I_{i}\right\}\ \mathrm{f}\!\!\!\mathrm{o}}\end{array}$$
m Xm Y m A m A fA θ f z ( ) () () () () ( ) = +⋅ ∑ ∑ 1 2 2  ,  and  or and  , (7)  non r CPRIM XY S XY AM z M XYS AN I I I I Φ ∈ Φ= ⊆ ⊆ Φ≠ ∈ ⊆ { } { } ,
or and  r XYS AN I I Φ≠ ∈ ⊆ { } , where ( ) f X is a function directly proportional to XfS , : 0, [ ] θ → ∞ . (8)  For example, 2 f () () X mX = I , or (9) 
$$(6)$$

$$(10)$$

$f(X)=\frac{card(X)}{card(M)}$ (ratio of cardinals), or $f(X)=m_{2\cap}(X)+card(X)$, etc.;
For example, $f(X)=m_{2\cap}(X)$, or $f(X)=card(X)$, or 
$${\mathrm{and~}}M{\mathrm{~is~a~subset~of~}}S^{\theta},{\mathrm{for~example:}}$$
and M is a subset of Sθ , for example: (10) 
where N is a subset of Sθ , for example: (11) 
$M$ is a subset of $X\bigcup Y$, etc., where $N$ is a subset of $S^{\theta}$, for example:  $N=X\bigcup Y$, or  $N$ is a subset of $X\bigcup Y$, etc.  
$$M=\left(X\cup Y\right),{\mathrm{or}}$$
$$\begin{array}{l}{{\tau\stackrel{\mathrm{~}}{\cup}\tau}}\\ {{\mathrm{~}\vdash S^{\theta}\,,\,\mathrm{for~example:}}}\\ {{M=\tau\left(X\cup Y\right),\,\mathrm{or}}}\end{array}$$
$$(11)$$
