X_i^{(1)}/X_i^{(0)},\, \ldots,\, X_i^{(k)} /X_i^{(0)})}$ has intensity measure $$\begin{aligned}
  \tilde \mu((c,\infty) \times [{\mathbf 0},{\mathbf s}]) 
  ={}& \mu\left(T^{-1}\left( (c,\infty) \times [{\mathbf 0},{\mathbf s}] \right) 
  \right)\\
  ={} & \mu(B_{\mathbf{s}} \cap ((c,\infty) \times [0,\infty)^k)) {}
  ={} c^{-1} \mu(B_{\mathbf{s}} \cap A)
 \end{aligned}$$ for any $c > 0$, $\mathbf{s} \in [0,\infty)^k$, where we use the fact that $\mu$, as an exponent measure, has the homogeneity property $c^{-1}\mu(\rd{\mathbf x})=\mu(\rd(c{\mathbf x}))$. Thus, $T\Pi$ has the same intensity as $\sum_{i\in{\mathbb N}} \delta_{(U_i, {\mathbf W}_i)}$, where $\sum_{i\in{\mathbb N}} \delta_{U_i}$ is a Fréchet point process and ${\mathbf W}_i$, $i \in {\mathbb N}$, are i.i.d. vectors with law $\sP({\mathbf W}\leq
   \mathbf{s}) = \mu(B_{\mathbf{s}} \cap A)$. Hence, we have $$\begin{aligned}
  \Xi\eqdist{}& \max_{i \in {\mathbb N}} T^{-1}\left(\big(X_i^{(0)}, X_i^{(1)} / X_i^{(0)}, 
  \ldots, X_i^{(k)} / X_i^{(0)}\big)\right)\\
  \eqdist{}& \max_{i \in {\mathbb N}} T^{-1}\left(\big(U_i,{\mathbf W}_i\big)\right) {}
  ={} \max_{i \in {\mathbb N}} U_i {\mathbf W}_i,
 \end{aligned}$$ which completes the proof.

\[ex:symm\_log\] For $T=\{t_0,\dots,t_k\}$, the symmetric logistic distribution is given by $$\begin{aligned}
    \sP(\xi(t_0) \leq x_0,\dots, \xi(t_k) \leq x_k) = 
    \exp\left[ - \left( x_0^{-q}+ \dots + x_k^{-q}\right)^{1/q} \right], 
    \label{eq:cdf_symm_log}
  \end{aligned}$$ for $x_0,\dots,x_k>0$ and $q > 1$. Hence, the density of the exponent measure is $$\begin{aligned}
    \mu(\rd x_0,\dots,\rd x_k) = 
    \left(\sum_{i=0}^k x_i^{-q}\right)^{1/q -(k+1)}
    \left(\prod_{i=1}^k(iq-1)\right)
    \prod_{i=0}^k x_i^{-q-1}  \rd x_0\dots \rd x_k.
  \end{aligned}$$ Applying 