**Proof.**

a. For *n* = 1, according to Theorem 1, since

$$GPHFWA\_{\lambda}(\widetilde{n}\_1) = \left(w\_1 \widetilde{n}\_1^{\lambda}\right)^{1/\lambda} = \left(\widetilde{n}\_1^{\lambda}\right)^{1/\lambda} = \widetilde{n}\_1.$$

Obviously, Equation (22) holds for *n* = 1.

b. For *n* = 2, since

$$
\begin{split}
\widetilde{m}\_{1} &= \mathop{\rm l}\_{\mathfrak{a}\_{1} \in \overline{\mathfrak{a}}\_{1}, \mathfrak{f}\_{1} \in \overline{\mathfrak{v}}\_{1}, \gamma\_{1} \in \overline{\mathfrak{v}}\_{1}} \left\{ \left\{ \mathfrak{a}\_{1}^{\lambda} \right\}, \left\{ 1 - \left( 1 - \beta\_{1} \right)^{\lambda} \right\}, \left\{ 1 - \left( 1 - \gamma\_{1} \right)^{\lambda} \right\} \right\}, \\
\widetilde{m}\_{2} &= \mathop{\rm l}\_{\mathfrak{a}\_{2} \in \overline{\mathfrak{v}}\_{2}, \beta\_{2} \in \overline{\mathfrak{v}}\_{2}, \gamma\_{2} \in \overline{\mathfrak{v}}\_{2}} \left\{ \left\{ \mathfrak{a}\_{2}^{\lambda} \right\}, \left\{ 1 - \left( 1 - \beta\_{2} \right)^{\lambda} \right\}, \left\{ 1 - \left( 1 - \gamma\_{2} \right)^{\lambda} \right\} \right\}.
\end{split}
$$

we have

$$\begin{split} w\_{1}\widetilde{\boldsymbol{n}}\_{1}^{\lambda} &= \underset{\boldsymbol{a}\_{1}\in\overline{\boldsymbol{\mu}}\_{1},\boldsymbol{\theta}\_{1}\in\overline{\boldsymbol{\eta}}\_{1},\boldsymbol{\gamma}\_{1}\in\overline{\boldsymbol{\nu}}\_{1}}{\operatorname{\boldsymbol{\mu}}} \left\{ \left\{ 1 - \left(1 - \boldsymbol{a}\_{1}\boldsymbol{\lambda}\right)^{\boldsymbol{w}\_{1}} \right\}\_{\boldsymbol{\cdot}}, \left\{ \left\{ 1 - \left(1 - \boldsymbol{\beta}\_{1}\right)^{\boldsymbol{\lambda}} \right\}^{\boldsymbol{w}\_{1}} \right\}\_{\boldsymbol{\cdot}}, \left\{ \left\{ 1 - \left(1 - \boldsymbol{\gamma}\_{1}\right)^{\boldsymbol{\lambda}} \right\}^{\boldsymbol{w}\_{1}} \right\}\_{\boldsymbol{\cdot}} \right\}, \\ w\_{2}\widetilde{\boldsymbol{n}}\_{2}^{\lambda} &= \underset{\boldsymbol{a}\_{2}\in\overline{\boldsymbol{\mu}}\_{2},\boldsymbol{\theta}\_{2}\in\overline{\boldsymbol{\eta}}\_{2},\boldsymbol{\gamma}\_{2}\in\overline{\boldsymbol{\nu}}\_{2}}{\operatorname{\boldsymbol{\mu}}} \left\{ \left\{ 1 - \left(1 - \boldsymbol{a}\_{2}\boldsymbol{\lambda}\right)^{\boldsymbol{w}\_{2}} \right\}^{\boldsymbol{w}\_{2}} \right\}\_{\boldsymbol{\cdot}} \left\{ \left\{ 1 - \left(1 - \boldsymbol{\beta}\_{2}\right)^{\boldsymbol{\lambda}} \right\}^{\boldsymbol{w}\_{2}} \right\}\_{\boldsymbol{\cdot}}, \left\{ \left\{ 1 - \left(1 - \boldsymbol{\gamma}\_{2}\right)^{\boldsymbol{\lambda}} \right\}^{\boldsymbol{w}\_{2}} \right\}\_{\boldsymbol{\cdot}}. \end{split}$$

then,

*w*1*n* 1 *λ* ⊕ *<sup>w</sup>*2*n*2*<sup>λ</sup>* = ∪ *<sup>α</sup>*1∈*μ* 1,*β*1∈*η*1,*γ*1∈*v*1,*α*2∈*μ*2,*β*2∈*η*2,*γ*2∈*v*2<sup>1</sup> − -1 − *<sup>α</sup>*1*<sup>λ</sup>*.*<sup>w</sup>*<sup>1</sup> + 1 − -1 − *<sup>α</sup>*2*<sup>λ</sup>*.*<sup>w</sup>*<sup>2</sup> − 1 − -1 − *<sup>α</sup>*1*<sup>λ</sup>*.*<sup>w</sup>*1<sup>1</sup> − -1 − *<sup>α</sup>*2*<sup>λ</sup>*.*<sup>w</sup>*2 , 1 − (1 − *<sup>β</sup>*1)*<sup>λ</sup><sup>w</sup>*1<sup>1</sup> − (1 − *<sup>β</sup>*2)*<sup>λ</sup><sup>w</sup>*2, 1 − (1 − *<sup>γ</sup>*1)*<sup>λ</sup><sup>w</sup>*1<sup>1</sup> − (1 − *<sup>γ</sup>*2)*<sup>λ</sup><sup>w</sup>*2. *w*1*n* 1 *λ* ⊕ *<sup>w</sup>*2*n*2*<sup>λ</sup>* = ∪ *<sup>α</sup>*1∈*μ* 1,*β*1∈*η*1,*γ*1∈*v*1,*α*2∈*μ*2,*β*2∈*η*2,*γ*2∈*v*2<sup>1</sup> − -1 − *<sup>α</sup>*1*<sup>λ</sup>*.*<sup>w</sup>*<sup>1</sup> -1 − *<sup>α</sup>*2*<sup>λ</sup>*.*<sup>w</sup>*2 , 1 − (1 − *<sup>β</sup>*1)*<sup>λ</sup><sup>w</sup>*1<sup>1</sup> − (1 − *<sup>β</sup>*2)*<sup>λ</sup><sup>w</sup>*2, 1 − (1 − *<sup>γ</sup>*1)*<sup>λ</sup><sup>w</sup>*1<sup>1</sup> − (1 − *<sup>γ</sup>*2)*<sup>λ</sup><sup>w</sup>*2.

and

$$\begin{array}{c} \text{GPPFF}(\overline{\mu}\_{1},\overline{\mu}\_{2}) - \left(\mathfrak{w}\_{1}\overline{\mathfrak{u}}\_{1}^{\lambda} \oplus \mathfrak{w}\_{2}\overline{\mathfrak{u}}\_{2}^{\lambda}\right)^{1/\lambda} = \\ \text{GPPFF}(\overline{\mu}\_{1},\overline{\mu}\_{2}) - \left\{\left\{\left(1-\left(1-\mathfrak{a}\_{1}^{\lambda}\right)^{\mathfrak{w}\_{1}}\left(1-\mathfrak{a}\_{2}^{\lambda}\right)^{\mathfrak{w}\_{2}}\right)^{1/\lambda}\right\}, \left\{1-\left(1-\left(1-\mathfrak{f}\_{1}\right)^{\lambda}\right)^{\mathfrak{w}\_{1}}\left(1-\left(1-\mathfrak{f}\_{2}\right)^{\lambda}\right)^{\mathfrak{w}\_{2}}\right\}^{1/\lambda}\right\}, \\ \text{GPPFF}(\overline{\mu}\_{1},\overline{\mu}\_{2}) - \left\{\left(1-\left(1-\left(1-\gamma\_{1}\right)^{\lambda}\right)^{\mathfrak{w}\_{1}}\left(1-\left(1-\gamma\_{2}\right)^{\lambda}\right)^{\mathfrak{w}\_{2}}\right)^{1/\lambda}\right\}. \end{array}$$

i.e., Equation (22) holds for *n* = 2.

$$\text{c.}\quad \text{If Equation (22) holds for } n = k\_\prime \text{ we have}$$

$$\begin{split} \text{GDPFWA}\_{\lambda}(\widetilde{\boldsymbol{\eta}}\_{1},\widetilde{\boldsymbol{\mu}}\_{2},\ldots,\widetilde{\boldsymbol{\mu}}\_{k}) &= \underset{\substack{\boldsymbol{\mu}\_{1}\in\widetilde{\boldsymbol{\mu}}\_{1},\boldsymbol{\mu}\_{2}\in\widetilde{\boldsymbol{\eta}}\_{2},\ldots,\boldsymbol{\mu}\_{k}\in\widetilde{\boldsymbol{\mu}}\_{k}}} \underset{\substack{\boldsymbol{\theta}\_{1}\in\widetilde{\boldsymbol{\eta}}\_{1},\boldsymbol{\theta}\_{2}\in\widetilde{\boldsymbol{\eta}}\_{2},\ldots,\boldsymbol{\theta}\_{k}\in\widetilde{\boldsymbol{\eta}}\_{k},\forall\varepsilon\in\mathbb{F}\_{k},\gamma\_{\varepsilon}\in\widetilde{\boldsymbol{\eta}}\_{2},\ldots,\gamma\_{k}\in\widetilde{\boldsymbol{\eta}}\_{k}} \left\{ \left\{ \left(1-\prod\_{j=1}^{k}\left(1-\boldsymbol{a}\_{j}^{A}\right)^{\boldsymbol{\eta}\_{j}} \right)^{1/\lambda} \right\},\ldots,\left(1-\prod\_{j=1}^{k}\left(1-\boldsymbol{a}\_{j}^{A}\right)^{\boldsymbol{\eta}\_{j}} \right) \right\}, \end{split}$$

$$\left\{ 1-\left(1-\prod\_{j=1}^{k}\left(1-\left(1-\left(1-\boldsymbol{\beta}\_{j}\right)^{\boldsymbol{\lambda}}\right)^{\boldsymbol{\eta}\_{j}}\right)^{1/\lambda} \right\}, \left\{ 1-\left(1-\prod\_{j=1}^{k}\left(1-\left(1-\gamma\_{j}\right)^{\boldsymbol{\lambda}}\right)^{\boldsymbol{\eta}\_{j}}\right)^{1/\lambda} \right\} \right\},$$

when *n* = *k* + 1, according to the operations of PHFEs, we have

*w*1*n* 1 *λ* ⊕ *<sup>w</sup>*2*n*2*<sup>λ</sup>* ⊕···⊕ *wknk<sup>λ</sup>* ⊕ *wk*+1*nk*+1*<sup>λ</sup>* = ∪ *<sup>α</sup>*1∈*μ* 1,*α*2∈*μ* 2,...,*αk*∈*μ k* ,*β*1∈*η*1,*β*2∈*η*2,...,*βk*∈*η<sup>k</sup>* ,*γ*1∈*v*1,*γ*2∈*v*2,...,*γk*∈*vk*33<sup>1</sup> − *k*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *<sup>α</sup>j<sup>λ</sup>*.*wj* 4 ,3 *k*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>β</sup>j*.*<sup>λ</sup>wj* 4, 3 *k*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>γ</sup>j*.*<sup>λ</sup>wj* 44 ⊕ ∪ *<sup>α</sup>k*+1∈*μk*+1,*β<sup>k</sup>*+1∈*ηk*+1,*γk*+1∈*vk*+<sup>1</sup><sup>1</sup> − -1 − *<sup>α</sup>k*+1*<sup>λ</sup>*.*wk*+<sup>1</sup>, 1 − (1 − *<sup>β</sup><sup>k</sup>*+<sup>1</sup>)*<sup>λ</sup>wk*+<sup>1</sup>, 1 − (1 − *<sup>γ</sup>k*+<sup>1</sup>)*<sup>λ</sup>wk*+<sup>1</sup> = ∪ *<sup>α</sup>*1∈*μ* 1,*α*2∈*μ* 2,...,*αk*+1∈*μ <sup>k</sup>*+1,*β*1∈*η*1,*β*2∈*η*2,...,*β<sup>k</sup>*+1∈*ηk*+1,*γ*1∈*v*1,*γ*2∈*v*2,...,*γk*+1∈*vk*+<sup>1</sup>33<sup>1</sup> − *k*+1 ∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *<sup>α</sup>j<sup>λ</sup>*.*wj* 4 ,3*k*+1 ∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>β</sup>j*.*<sup>λ</sup>wj* 4, 3*k*+1 ∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>γ</sup>j*.*<sup>λ</sup>wj* 44.

then,

$$\begin{split} \operatorname{GPFFNA}\_{\lambda}(\mathring{\mathfrak{h}}\_{1},\mathring{\mathfrak{h}}\_{2},\ldots,\mathring{\mathfrak{h}}\_{k}) &= \left(\operatorname{w}\_{1}\mathring{\mathfrak{h}}\_{1}^{\lambda}\otimes\operatorname{w}\_{2}\mathring{\mathfrak{h}}\_{1}^{\lambda}\otimes\cdots\otimes\operatorname{w}\_{k}\mathring{\mathfrak{h}}\_{k}^{\lambda}\otimes\operatorname{w}\_{1+\frac{1}{2}}\mathring{\mathfrak{h}}\_{1+\frac{1}{2}}\right)^{1/\lambda} \\ &\times \operatorname{w}\_{1}\mathring{\mathfrak{v}}\_{1}\star\mathring{\mathfrak{v}}\_{2}\psi\_{2}\ldots\psi\_{k+1}\psi\_{k+1}\mathring{\mathfrak{v}}\_{1}\wedge\mathring{\mathfrak{v}}\_{2}\psi\_{2}\ldots\otimes\operatorname{w}\_{k+1}\mathring{\mathfrak{v}}\_{1+1}\psi\_{1}\psi\_{1}\psi\_{1}\psi\_{2}\psi\_{2}\ldots\psi\_{k+1}\psi\_{k+1} \\ &\times \left\{1\left(1-\prod\_{j=1}^{k+1}\left(1-\left(1-\gamma\_{j}\right)^{\lambda}\right)^{\mathsf{T}}\right)^{1/\lambda}\right\}, \Bigg{1\geqslant \\ &\left\{1-\left(1-\prod\_{j=1}^{k+1}\left(1-\left(1-\gamma\_{j}\right)^{\lambda}\right)^{\mathsf{T}}\right)^{1/\lambda}\right\}. \end{split}$$

i.e., Equation (22) holds for *n* = *k* + 1; we can demonstrate that Equation (22) holds for all values of *n*. -
