= Lifting a subspace through a Freiman model
{title2=$W\subseteq4L(B)\Longrightarrow \widetilde W\subseteq4B$}
Under <injective linear modelling of a small sumset>, every $w\in W$ has a unique lift $\sigma(w)\in4B$, because differences of two lifts lie in $8B\subseteq12B$. For $w_1,w_2\in W$, the discrepancy $\sigma(w_1)+\sigma(w_2)+\sigma(w_1+w_2)$ lies in $12B\cap\ker L$, so it is zero. Thus $\sigma$ is a <linear map> and its image is a <vector subspace> of the same size as $W$. This verifies the additive lifting step rather than assuming that a preimage chosen from a sumset is automatically a subspace.
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