theorem is the following.

\[lifting thm convenient\] Let $D = \sum_{i=1}^k a_i - \sum_{j = 1}^k b_j$ be a principal divisor on $\Gamma$. Assume that $\sum_{i=1}^g a_i$ is a break divisor supported on two valent points. Then, given preimages $x_i$ and $y_j$ for all $i = g+1,\dots,k$ and all $j = 1,\dots,k$ such that $\tau(x_i) = a_i$ and $\tau(y_j) = b_j$, there exist $x_1,\dots,x_g \in X(K)$ with $\tau(x_i) = a_i$ such that $\sum_{i=1}^k x_i - \sum_{j=1}^k  y_i$ is a principal divisor on $X$.

This follows from the lifting theorem applied with $$D = \sum_{i = 1}^{k} b_i - \sum_{i = g+1}^k a_i \text{ and }  B = \sum_{i=1}^{g} x_i.
\qedhere$$

Constructions of the piecewise-linear functions and lifting
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We construct the piecewise-linear functions $F_1$, $F_2$ and $F_3$ by specifying their divisors. To construct these divisors, we will need to choose, for each edge $e$, points which will be labeled $c_e, a_e, p_e, q_e, b_e, d_e$ in the interior of $e$. This will be the order of the points in their respective edge. We also require that the pairs $c_e, d_e$ and $a_e, b_e$ and $p_e, q_e$ are symmetric about the middle of their edges.

We will describe the exact position 