Spherical Separation Theorem

In this paper, it is shown that for any two non-empty closed (resp., open) and spherical convex subsets ${\mathcal{W}}_1,{\mathcal{W}}_2$ of $S^n$, the intersection ${\mathcal{W}}_1\cap {\mathcal{W}}_2$ is empty if and only if the subset ${P\in S^n; |; P\cdot Q>0 \mbox{ for any } Q\in $\mathcal{W}$_1 \mbox{ and } P\cdot R<0 \mbox{ for any } R\in $\mathcal{W}$_2}$ is non-empty, open (resp., closed) and spherical convex.