It is established the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an

${\mathcal{U}}_{c}^{K}$-multimap or a a multimap

$T:A\to {2}^{B}$ such that both

$T$ and

$S\circ T$ are closed and have the KKM property for each Kakutani multimap

$S:B\to {2}^{B}$. As an application, existence theorems of equilibrium pairs for free

$n$-person games as well as for free 1-person games are proved.