**Appendix B**

**Proof.** According to Theorem 2, since *nj* = *n* = {*μ*, *η*, *<sup>v</sup>*}, we have

$$\begin{split} \underset{\boldsymbol{\tilde{u}},\boldsymbol{\tilde{v}},\boldsymbol{\tilde{p}},\boldsymbol{\tilde{\theta}} \in \overline{\operatorname{\mathbb{R}}},\boldsymbol{\tilde{v}},\boldsymbol{\tilde{v}} \in \overline{\operatorname{\mathbb{R}}}}{\operatorname{\mathbb{\mathbb{Q}}}} \left\{ \left\{ \left(1 - \prod\_{j=1}^{n} (1 - \boldsymbol{a}^{\lambda})^{\boldsymbol{w}} \right)^{1/\lambda} \right\}, \left\{ \left(1 - \prod\_{j=1}^{n} \left(1 - (1-\boldsymbol{\beta})^{\boldsymbol{\lambda}} \right)^{\boldsymbol{w}} \right)^{1/\lambda} \right\}, \left\{ 1 - \left(1 - \prod\_{j=1}^{n} \left(1 - (1-\boldsymbol{\gamma})^{\boldsymbol{\lambda}} \right)^{\boldsymbol{w}} \right)^{1/\lambda} \right\} \right\} \\ = \underset{\boldsymbol{\tilde{u}},\boldsymbol{\tilde{v}},\boldsymbol{\tilde{b}} \in \overline{\operatorname{\mathbb{R}}},\boldsymbol{\tilde{v}} \in \overline{\operatorname{\mathbb{R}}}}{\operatorname{\mathbb{R}}} \left\{ \left\{ \left(1 - \left(1 - \boldsymbol{a}^{\lambda} \right) \right)^{1/\lambda} \right\}, \left\{ 1 - \left(1 - \left(1 - (1-\boldsymbol{\beta})^{\boldsymbol{\lambda}} \right) \right)^{1/\lambda} \right\}, \left\{ 1 - \left(1 - \left(1 - (1-\boldsymbol{\gamma})^{\boldsymbol{\lambda}} \right) \right)^{1/\lambda} \right\} \right\} \\ = \underset{\boldsymbol{\tilde{u}},\boldsymbol{\tilde{v}},\boldsymbol{\tilde{b}} \in \overline{\operatorname{\mathbb{R}}},\boldsymbol{\tilde{v}} \in \overline{\operatorname{\mathbb{R}}},\boldsymbol{\tilde{v}} \in \overline{\operatorname{\mathbb{R}}},\boldsymbol{\tilde{v}}$$

