The proof of Proposition 2.13 also gives that if $\varphi\colon V\to{\mathbb{R}}$ is a piecewise ${\mathbb{Q}}$-linear function on a paracompact strictly $K$-analytic space $V$, then there exists a family $(\varphi_{i})_{i\in I}$ of piecewise ${\mathbb{Q}}$-linear functions on $V$ such that the family ${\rm supp}(\varphi_{i})_{i\in I}$ is a locally finite family of compact sets subordinate to any given open covering of $V$ and such that $\varphi=\sum_{i\in I}\varphi_{i}$. Indeed, in the above proof we may construct the covering $U_{i}$ finer than the given open covering and then we may use $\varepsilon=0$ in the construction due to piecewise ${\mathbb{Q}}$-linearity.