With BCR17, for D2, m(θ1) = 0.2 is transferred proportionally to all elements of D1, i.e. xθ2 0.1 =
yθ3 0.2 =
zθ2∪θ3 0.1 =
0.2 0.4 = 0.5 whence the parts of m(θ1) redistributed to θ2, θ3 and θ2 ∪ θ3 are respectively xθ2 = 0.05, yθ3 = 0.10, and zθ2∪θ3 = 0.05. For D3, there is actually no need to transfer m(θ1∪θ3) because m(θ1∪θ3) = 0 in this example; whereas m(θ1∪θ2) = 0.1 is transferred to θ2 (no case of k-elements herein); m(θ1∪θ2∪θ3) =
0.3 is transferred to θ2, θ3 and θ2 ∪ θ3 proportionally to their corresponding masses:

$$x_{\theta_{2}}/0.1=y_{\theta_{3}}/0.2=z_{\theta_{2}\cup\theta_{3}}/0.1=0.3/0.4=0.75$$  whence $x_{\theta_{2}}=0.075$, $y_{\theta_{3}}=0.15$, and $z_{\theta_{2}\cup\theta_{3}}=0.075$. Finally, one gets  $$m_{BC17}(\theta_{2}|\theta_{2}\cup\theta_{3})=0.10+0.05+0.10+0.075=0.325$$ $$m_{BC17}(\theta_{3}|\theta_{2}\cup\theta_{3})=0.20+0.10+0.15=0.450$$ $$m_{BC17}(\theta_{2}\cup\theta_{3}|\theta_{2}\cup\theta_{3})=0.10+0.05+0.075=0.225$$

which is different from the result obtained with SCR, since one gets in this example:

$$\begin{array}{l}{{m_{S C R}(\theta_{2}|\theta_{2}\cup\theta_{3})=m_{S C R}(\theta_{3}|\theta_{2}\cup\theta_{3})=0.25}}\\ {{m_{S C R}(\theta_{2}\cup\theta_{3}|\theta_{2}\cup\theta_{3})=0.50}}\end{array}$$

More complex and detailed examples can be found in [34].

## 7.4 Qualitative Belief Conditioning Rules

In this section we present only the qualitative belief conditioning rule no 17 which extends the principles of the previous quantitative rule BCR17 in the qualitative domain using the operators on linguistic labels defined previously. We consider from now on a general frame Θ = {θ1, θ2**, . . . , θ**n}, a given model M(Θ) with its hyperpower set DΘ and a given extended ordered set L of qualitative values L = {L0, L1, L2, . . . , Lm, Lm+1}. The prior qualitative basic belief assignment (qbba) taking its values in L is denoted qm(.). We assume in the sequel that the conditioning event is A 6= ∅, A ∈ DΘ, i.e. the absolute truth is in A. The approach we present here is a direct extension of BCR17 using FLARL operators. Such extension can be done with all quantitative BCR's rules proposed in [34], but only QBCR17 is presented here for the sake of space limitations. 7.4.1 Qualitative Belief Conditioning Rule no 17 (QBCR17)
Similarly to BCR17, QBCR17 is defined by the following formula:

$$qm_{BCR1T}(X|A)=qm(X)\cdot\left[qS_{D_{1}}+\sum_{\begin{subarray}{c}W\in D_{2}\cup D_{3}\\ X\subset W\end{subarray}}\frac{qm(W)}{qS(W)}\right]+\sum_{\begin{subarray}{c}W\in D_{2}\cup D_{3}\\ X\subset W,X\text{is}k\text{-largest}\\ qS(W)=0\end{subarray}}qm(W)/k\tag{36}$$

where "X is k-largest" means that X is the k-largest (with respect to inclusion) set included in W and

$qS(W)\stackrel{{\Delta}}{{=}}\sum_{Y\in D_{1},Y\subset W}qm(Y)$  $\sum_{Z\in D_{1},}qm(Z)$  $S_{D_{1}}\stackrel{{\Delta}}{{=}}\frac{\text{or}Z\in D_{2}\mid\frac{4Y\in D_{1}\text{with}Y\subset Z}{\sum_{Y\in D_{1}}qm(Y)}}{\sum_{Y\in D_{1}}qm(Y)}$  $\sum_{Y\in D_{1}}qm(Y)$ 
Naturally, all operators (summation, product, division, etc) involved in the formula (36) are the operators defined in FLARL working on linguistic labels. It is worth to note that the formula (36) requires also the division of the label qm(W) by a scalar k. This division is defined as follows:
Let r ∈ R, r 6= 0. Then the label division by a scalar is defined by La r= La/r (37)