Proof of Theorem 1:

We have \((\wp_ }} )^ }} = \left\langle \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right.\), \(\left. \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle .\)

$$\therefore \,\,\mathop \otimes \limits_^ (\wp_ }} )^ }} = \left\langle ^ \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right.,$$

$$\left. ^ \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin =&\; \left\langle ^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}} \right., \\& \left. ^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}} \right\rangle. \end$$

Next, we have:

$$\begin & \mathop \oplus \limits_ < p_ < \cdots < p_ \le n}} \left( ^ (\wp_ }} )^ }} } \right) \\&\quad = \left\langle < p_ < ... < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right. \end$$

$$\, \left. < p_ < ... < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}^ \left( }}^ }} }}^ }}} \right)^ } } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin = &\; \left\langle < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right., \\& \left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right\rangle . \end$$

$$\begin & \therefore \,\,\,\frac^c_ }}\mathop \oplus \limits_ < p_ < \cdots < p_ \le n}} \left( ^ (\wp_ }} )^ }} } \right) \\& \quad = \left\langle ^C_ }}\left( < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }} < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right., \end$$

$$\, \left. ^C_ }}\left( < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }} < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin =&\; \left\langle ^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}} \right., \\& \left. ^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}} \right\rangle . \end$$

Hence,

$$FFDGMSM(\wp_ ,\;\wp_ ,\;...,\;\wp_ ) = \left( ^c_ }}\mathop \oplus \limits_ < p_ < \cdots < p_ \le n}} \left( ^ (\wp_ }} )^ }} } \right)} \right)^ + t_ + \cdots + t_ }}}}$$

$$= \left\langle + t_ + \cdots + t_ }}\left( ^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right.,$$

$$\, \left. + t_ + \cdots + t_ }}\left( ^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}^C_ }}\left. < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)} \right)} \right)} \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle$$

$$= \left\langle + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right.,$$

$$\,\,\,\,\left. + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }}^ }} }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right\rangle .$$

Proof of Theorem 2:

$$\begin FFDGMSM(\wp_ ,\;\wp_ ,\;...,\;\wp_ ) \hfill \\ = \left\langle + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( ^ }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right., \hfill \\ \,\,\,\,\left. + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( ^ }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right\rangle \hfill \\ \end$$

$$\begin = \left\langle + t_ + \cdots + t_ }}\left( ^C_ }} \times (t_ + t_ + \cdots + t_ )^ \times ^C_ \times \left( ^ }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right., \hfill \\ \,\,\,\,\left. + t_ + \cdots + t_ }}\left( ^C_ }} \times (t_ + t_ + \cdots + t_ )^ \times ^C_ \times \left( ^ }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right\rangle \hfill \\ \end$$

$$= \left\langle ^ }}^ }}} \right\}} \right)^ }}} \right.,\left. ^ }}^ }}} \right\}} \right)^ }}} \right\rangle = \left\langle ^} ,\delta_^} } \right\rangle = \wp_ .$$

Proof of Theorem 3:

$$\begin \mathop \limits_ \varphi_^} \le \varphi_^} \le \mathop \limits_ \varphi_^} \hfill \\ \Rightarrow 1 - 1 - (\mathop \limits_ \varphi_^} )^ \ge 1 - \varphi_^ \ge 1 - (\mathop \limits_ \varphi_^} )^ \hfill \\ \Rightarrow \sum\limits_^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } \ge \sum\limits_^ \left( ^ }}^ }}} \right)^ } \ge \sum\limits_^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } \hfill \\ \Rightarrow \left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } } \right)^ } } \right)^ \hfill \\ \,\,\,\,\,\, \ge \left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( ^ }}^ }}} \right)^ } } \right)^ } } \right)^ \hfill \\ \,\,\,\,\,\, \ge \left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } } \right)^ } } \right)^ \hfill \\ \end$$

$$\begin \Rightarrow \sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \,\,\,\,\,\, \le \sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( ^ }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \,\,\,\,\, \le \sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \varphi_^} )^ }}\limits_ \varphi_^} )^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \end$$

Similarly, it can be shown that

$$\begin \,\,\sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \delta_^} )^ }}\limits_ \delta_^} )^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \,\,\, \ge \sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( ^ }}^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \,\, \ge \,\sqrt[3] + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( \limits_ \delta_^} )^ }}\limits_ \delta_^} )^ }}} \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }} \hfill \\ \end$$

Hence \(\wp^ \prec FFDGMSM(\wp_ ,\;\wp_ ,\;...,\;\wp_ ) \prec \wp^\).

Proof of Theorem 5:

We have

$$\begin w_ }} \wp_ }} =&\; \left\langle }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right.,\\& \,\left. }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle. \end$$

$$\therefore \,\,(w_ }} \wp_ }} )^ }} = \left\langle \left( }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }} }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right.,\,$$

$$\, \left. \left( }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }} }} \left( }}^ }} }}^ }}} \right)^ } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin = &\; \left\langle \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right.,\\& \left. \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right\rangle . \end$$

$$\begin & \therefore \,\,\mathop \otimes \limits_^ (w_ }} \wp_ }} )^ }} \\& \quad = \left\langle ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }} \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right., \end$$

$$\, \left. ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }} \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin =&\; \left\langle ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}} \right.,\\& \left. ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}} \right\rangle . \end$$

Next, we have:

$$\begin & \mathop \oplus \limits_ < p_ < \cdots < p_ \le n}} \left( ^ (w_ }} \wp_ }} )^ }} } \right) \\&\quad = \left\langle < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right., \end$$

$$\,\,\,\,\left. < p_ < ... < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right\}^} } \right)^ }}} \right)^ } } \right\}^} } \right)^ }}} \right\rangle$$

$$\begin =&\; \left\langle < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right.,\\& \left. < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right\rangle . \end$$

$$\begin & \therefore \,\,\,\frac^c_ }}\mathop \oplus \limits_ < p_ < \cdots < p_ \le n}} \left( ^ (w_ }} \wp_ }} )^ }} } \right) \\& \quad = \left\langle ^C_ }}\left( < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }} < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right., \end$$

$$\,\,\, \, \,\,\,\,\left. ^C_ }}\left( < p_ < ... < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }} < p_ < ... < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle .$$

$$\begin = &\; \left\langle ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right.,\\& \left. ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right\rangle . \end$$

Hence, \(FFWDGMSM(\wp_ ,\;\wp_ ,\;...,\;\wp_ )\)

$$= \left\langle + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right.,$$

$$\left. + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right\}^} } \right)^ }}} \right)^ } \right\}^} } \right)^ }}} \right\rangle$$

$$= \left\langle + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right.,$$

$$\,\,\,\,\,\left. + t_ + \cdots + t_ }}\left( ^C_ }}\sum\limits_ < p_ < \cdots < p_ \le n}} ^ \left( }} \left( }}^ }} }}^ }}} \right)^ } \right)^ } } \right)^ } } \right)^ } \right\}^} } \right)^ }}} \right\rangle .$$