and an array $(\mathcal{U}(T_0,\cdots,T_k):k<\omega)$ where

1.  [For each $k$, $N_k$ is a *symmetric*[^3] neighborhood of the identity of $G$;]{}

2.  [With $n_0$ such that $0\in S_{n_0}$, $\mathcal{U}(\emptyset)$ is a finite subset of $F(\emptyset)$ (ONE’s first move) which covers $G_0$, and $N_0$ is a neighborhood of the identity of $G$ such that for each $x\in G_0$ there is a $V\in\mathcal{U}(\emptyset)$ with $x*(N_0*N_0)\subseteq V$.]{}

3.  [For each $(T_0,\cdots,T_k)$ such that $T_0\in F(\emptyset)$, $T_1\in F(T_0)$, $\cdots$ and $T_k\in F(T_0,\cdots,T_{k-1})$ and for $n_{k+1}$ such that $k+1\in S_{n_{k+1}}$ we have $\mathcal{U}(T_0,\cdots,T_k)$ a finite subset of $F(T_0,\cdots,T_k)$ that covers $G_{n_{k+1}}$. Note that there are only finitely many such $(T_0,\cdots,T_k)$. $N_{k+1}$ is a neighborhood of the identity of $G$ such that for each such sequence $(T_0,\cdots,T_k)$ and for each $x\in G_{n_{k+1}}$ there is a $U\in\mathcal{U}(T_0,\cdots,T_{k})$ with $x*N_{k+1}*N_{k+1}\subseteq U$.]{}

With this data available, construct a play against $F$ won by TWO as follows: Fix an $m<\omega$. Since $X_m$ is Rothberger bounded select for each $k\in S_m$ an $x_k\in G$ such that $(x_k*N_k:k\in S_m)$ covers $X_m$.

We may assume each $x_k$ is in $X_m$ - for suppose an $x_k$ is not in $X_m$. If $x_k*N_k \cap X_k = \emptyset$, we may with impunity replace this $x_k$ by one from 