In this paper, it's proven in 1.7 that
If $V$ is an analytic subset of dimension $\leq q$ in an open neighborhood $U$ of $x_0$ in $D$, then there exists an open neighborhood $U'$ of $x_0$ in $U$ such that $E_c\cap V\cap U'$ is an analytic subset of $V\cap U'$.
The goal is to prove that $E_c$ is an analytic subset of $D$.
Applying the above to the trivial analytic subset $V=D$, one obtains that $E_c$ is locally analytic (i.e. $\forall x_0\in D$, there exists $U'$ such that $E_c\cap U'$ is an analytic subset of $U'$). Since there is no more explanation, it should be straightforward to deduce that $E_c$ is an analytic subset of $D$. However I couldn't do that.
How does one deduce that $E_c$ is an analytic subset of $D$?