Let $\varphi\in\mathcal{D}(X)$. By Proposition 5.2 we may find a model $\mathcal{X}$ and a model function $\psi$ such that $\theta$, $\varphi$ and $\psi$ are all determined on $\mathcal{X}$ and such that $\theta+dd^{c}\psi$ is $\mathcal{X}$-positive. Since the closed $(1,1)$-form $dd^{c}\varphi$ is determined on $\mathcal{X}$ we may thus find a rational number $0<\varepsilon\ll 1$ such that $\theta+dd^{c}(\psi+\varepsilon\varphi)\geq 0$. It follows that $\varepsilon\varphi=(\psi+\varepsilon\varphi)-\psi$ is a difference of $\theta$-psh model functions, and the result follows. The case when $\theta$ is semipositive is proved in a similar way. ∎