 an $\mathcal{A}$-ample set $\Pi$ of places for $\varphi$ such that $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}| + 0^{k}$, for some promising set $\mathcal{A}$ of atoms in $\varphi$, and a map $x \mapsto \pi^{x}$ from $\mathsf{V}_{0}$ into $\Pi$ such that $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for $x \in \mathsf{V}_{0}$. 