= Solution
Because $A\subseteq\mathbb Z^2$ is finite, choose $R$ with $A\subseteq[-R,R]^2$ and an integer $M>2sR$. Define
$$
\phi(x,y)=x+My.
$$
An equality of $s$-term sums in $\mathbb Z^2$ plainly gives equality after applying this <linear map>. Conversely, equality of the images gives
$$
\Delta_x+M\Delta_y=0,
$$
where $|\Delta_x|\leq2sR<M$. Therefore $\Delta_x=0$, and then $\Delta_y=0$. The same estimate with one term on each side shows that $\phi$ is injective on $A$. Hence $\phi$ is a <Freiman s-isomorphism> from $A$ to the subset $\phi(A)\subseteq\mathbb Z$.
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