 that $S{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {};}}+ B$ is a ${\mathcal{P}}$-position by proving that for every first player's move, the second player always has an answer leading to a ${\mathcal{P}}$-position. We use induction on the size of $S$.  The cases where $S$ is empty