The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} \mathscr{P}_{F}(A)$ such that ${}^{\ast}a \in B$ for each $a \in A$. ($\mathscr{P}_{F}(A)$ is the set of all finite subsets of $A$)
A binary relation $P$ is concurrent on $A \subseteq \operatorname{dom}{P}$ if for each finite set $\{x_i\}_{1 \leq i \leq n}$ in $A$ there is a $y \in \operatorname{range}{P}$ so that $\langle x_i, y\rangle \in P$ for $1 \leq i \leq n$.$P$ is concurrent, if it is concurrent on $\operatorname{dom}{P}$.
How to show if that for each concurrent relation $P \in V(X)$ there is an element $b \in \operatorname{range}{{}^{\ast}P}$ so that $\langle {}^{\ast}x, b\rangle \in{}^{\ast}P$ for all $x \in\operatorname{dom}{P}$, then $V({}^{\ast}X)$ is an enlargement of $V(X)$?