for Theorem \[theorem-identification\] {#proof-identification}
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The identifiability of the model and is equivalent to that as long as $\Omega\not=\Omega_0$, $$\label{unique-maximum}
{\scalebox{0.9}{$
\begin{aligned}
&p_{a}\left(g\left(z,t,t\mid a\right),0\right)\mathcal{J}_{z|a}(t)\exp(-\int_{0}^{t}\exp(b^Tg\left(z,t,s\mid a\right))\lambda(t-s)ds)\lambda(t)\exp\left(b^{T}z\right)\\
&\not=
p_{a_0}\left(g\left(z,t,t\mid a_0\right),0\right)\mathcal{J}_{z|a_0}(t)\exp(-\int_{0}^{t}\exp(b_0^{T}g\left(z,t,s\mid a_0\right))\lambda_0(t-s)ds)\lambda_0(t)\exp\left(b_0^{T}z\right)
\end{aligned}$}}$$ within a positive measure set $M\in\mathbb{R}^{p}\times\mathbb{R}_{+}$ (with respect to the standard Lebesgue Measure).

\(1) Suppose does not hold for some $\Omega$ with $b\not=b_0$. The right-continuity condition in $C1$ requires that $$\label{initial identity}
p_a(z,0)\exp(b^Tz)\equiv p(z,0)\exp(b_0^Tz)$$ where $p(.,0)$ is the true initial pdf. The assumption that the true joint pdf has the full support implies that $p(.,0)$ has full support as well. Therefore, Eq. leads to $$p_a(z,0)\equiv p(z,0)\exp\left(\left(b_0-b\right)^Tz\right),$$ both $p_a(.0)$ and $p(.,0)$ are probability density function, which yield that $$\int_{\mathbb{R}^{p}}p(z,0)\exp(c\cdot z)dz=1$$ for some $c\not=0$ that contradicts to the requirement in $C6$. Consequently, every $\Omega$ that could potentially break down the condition must have $b=b_0$.

\(2) On the other hand, if $b=b_0$, we have for all $t\geq 0$: $$\label{condition-for-lambda}
S_{(a,b_0,\lambda)}(t)\cdot \lambda(t)=S_{(a_0,b_0,\lambda_0)}(t)\cdot \lambda_0(t)$$ where $S_{(a,b,\lambda)}$ is the survival function of the failure time that follows the joint pdf associated with $\Omega=(a,b,\lambda)$, by definition it has the following form: $${\scalebox{0.9}{$
S_{(a,b,\lambda)}(t):\int_{\mathbb{R}^p}p_{a}\left(g\left(z,t,t\mid a\right),0\right)\mathcal{J}_{z|a}(t)\exp(-\int_{0}^{t}\exp(b_0^Tg\left(z,t,s\mid a\right))\lambda(t-s)ds)dz\lambda(t)$}}.$$ Because the survival function is uniquely determined by the pdf of the event time which is furthermore uniquely determined by the joint 