Let's denote $m_{2\cap}(A)=\big{(}m_{1}\oplus m_{2}\big{)}(A)=\sum_{\begin{subarray}{c}X\subseteq X^{\text{\tiny{\textregistered}}}\\ (X\cap Y)=A\end{subarray}}m_{1}(X)m_{2}(Y)$ using the conjunction rule.  
 Let's note the set of intersections by: 

$$S_{\cap_{1}}=\begin{cases}X\in S^{\theta}\mid X=y\bigcap z,\text{where}y,z\in S^{\theta}\setminus\{\Phi\},\\ X\text{is in a canonical form,and}\\ X\text{contains at least an}\bigcap\text{symbol in its formula}\end{cases}.\tag{1}$$

In conclusion, S1 is a set of formulas formed with singletons (elements from the frame of discernment), such that each formula contains at least an intersection symbol 1, and each formula is in a canonical form (easiest form). 

 For example: AAS I ∉ I since A A I is not a canonical form, and AAA I = . Also, 
( ) *AB B* I I is not in a canonical form but ( *AB B ABS* **II I** ) = ∈ I . 

 Let S ΦI = the set of all empty intersections from SI , 
 and 
,
non r S Φ
I = {the set of all non-empty intersections from non S Φ
I whose masses are redistributed to other sets, which actually depends on the sub-model of each application}. 

## 2. Extension Of Inagaki General Weighted Operators (Wo).

Inagaki general weighted operator ( ) WO is defined for two sources as: 
$$\forall\;A\in\;2^{\theta}\setminus\left\{\Phi\right\},\;m_{(W O)}(A)=\sum_{\begin{subarray}{c}{{X,Y\in\mathbb{Z}^{\theta}}}\\ {{(X\cap Y)=A}}\end{subarray}}m_{1}(X)m_{2}(Y)+W_{m}(A)\cdot m_{2\cap1}(\Phi)\,,$$
$$\quad(2)$$
$$\left({\boldsymbol{3}}\right)$$
where 

$\sum_{m}(X)=1$ and all $W_{m}(\cdot)\in[0,1]$.  $X\in2^{\theta}$  : conflicting mass is redistributed to 
∑ = and all Wm ( ) 0,1 ⋅ ∈[ ]. (3) 
 So, the conflicting mass is redistributed to non-empty sets according to these weights 
( ) Wm ⋅ . 

 In the extension of this WO , which we call the Double Weighted Operator ( ) DWO , we redistribute not only the conflicting mass 2 m I ( ) Φ but also the mass of some (or all) non-empty intersections, i.e. those from the set ,
non r S Φ
I , to non-empty sets from Sθ according to some weights 
( ) Wm ⋅ for the conflicting mass (as in WO), and respectively according to the weights Vm(.) for the non-conflicting mass of the elements from the set ,
non r S Φ
I : 

$$\forall\;A\in\;\left(S^{\phi}\setminus S_{\Omega,r}^{m\phi}\right)\setminus\left\{\Phi\right\},\;m_{D m\omega}(A)=\sum_{\begin{subarray}{c}X,Y\in S^{\phi}\\ (X|Y)=A\end{subarray}}m_{1}(X)m_{2}(Y)+W_{m}(A)\cdot m_{221}(\Phi)+V_{m}(A)\cdot\sum_{z\in\mathbb{R}_{\omega}^{m\phi}}m_{221}(z)\,,$$

, 
2