)$ so $S{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {};}}+B$ is a ${\mathcal{P}}$-position.  -   $S=S_{\ell}$ with $\ell \geq 0$ or $S=S_{0,\ell}$ with $\ell \geq 0$. By Observation \[lem:keylemma\], the second player will always be able to replicate the first player's move on $B$, by playing the symmetrical move. By induction hypothesis, the new position is a ${\mathcal{P}}$-position.  Figure \[fig:P0