a function $\tilde{f}(w_1, \cdots, w_{2n+m}) = 
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} 
&=& \frac{1}{2} \sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}} \nonumber \\
&&+ \sum_{a=1}^{2n}\sum_{s=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(m)}\end{aligned}$$ where $\tilde{H}= \tilde{H}(w_1, \cdots, w_{2n+m}) 
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$. Introducing functions $\tilde{G}_s$ $(s=1, \cdots, m)$ and $\tilde{g}^{(m)}_{ab}$ that satisfy the following relations, $$\begin{aligned}
\frac{1}{2} \{x_a, x_b\}_{\mbox{\tiny{PB}}}
&=& 
\tilde{g}^{(m)}_{ab} \frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{m})}~,~~
\label{phi-2n+m1}\\
\frac{1}{2}\{x_a, z_{s}\}_{\mbox{\tiny{PB}}}
&=& 
-\sum_{b=1}^{2n} \tilde{g}^{(m)}_{ab} 
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{s-1},
\tilde{G}_{s}, \tilde{G}_{s+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_b, z_{s+1}, \cdots,z_{m})}~,~~
\label{phi-2n+m2}\\
\{z_s, z_t\}_{\mbox{\tiny{PB}}}
&=&\!\!\!\!
\sum_{a,b=1}^{2n}\tilde{g}^{(m)}_{ab} 
\frac{\partial (\tilde{G}_1, \cdots, 
\tilde{G}_{s-1}, \tilde{G}_{s}, \tilde{G}_{s+1}, \cdots, 
\tilde{G}_{t-1}, \tilde{G}_{t}, \tilde{G}_{t+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_{a}, z_{s+1},
\cdots, z_{t-1}, x_{b}, z_{t+1}, \cdots, z_{m})}~,
\label{phi-2n+m3}\end{aligned}$$ where $s<t$, Eq. (\[H-eq(m)\]) can be rewritten as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{a, b=1}^{2n} \tilde{g}^{(m)}_{ab} 
\frac{\partial (\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)}
\label{N-eqf(m)}\end{aligned}$$ using a formula concerning Jacobians.

By the use of Eqs. (\[phi-2n+m1\])–(\[phi-2n+m3\]), it is shown that the Poisson bracket between any of the $m$ functions $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{G}_s(w_1, \cdots, w_{2n+m})$ 