with our lifting theorem. Notice for example the slight bumps in Figure \[fig:F2-graphT\], which are there specifically to allow application of our lifting theorem.

![The graph of $F_2|_{e}$ for $e \notin T$.[]{data-label="fig:F2-graphT"}](F1){width=".8\linewidth"}

![The graph of $F_2|_{e}$ for $e \notin T$.[]{data-label="fig:F2-graphT"}](F3){width=".8\linewidth"}

![The graph of $F_2|_{e}$ for $e \notin T$.[]{data-label="fig:F2-graphT"}](F2){width=".8\linewidth"}

![The graph of $F_2|_{e}$ for $e \notin T$.[]{data-label="fig:F2-graphT"}](F2ei){width=".8\linewidth"}

We now want to lift these functions to ${X^{\rm an}}$ by lifting their divisors using Theorem \[lifting thm convenient\].

\[D3 lift\] For every $e$ there exist lifts $a'_e, b'_e \in X(K)$ of $a_e, b_e$ such that $$\begin{aligned}
D'_3 \coloneqq \sum_{e \in E} a'_e - b'_e\end{aligned}$$ is a principal divisor on X.

\[D1 lift\] For every point in $\{ v(e), w(e), p_e, q_e \mid e \in E \}$ there exist a lift in $X(K)$, which we denote by $v(e)', w(e)', p'_e, q'_e$ respectively such that $$\begin{aligned}
D'_1 \coloneqq \sum_{e \in E} v(e)' + w(e)' - p'_e - q'_e\end{aligned}$$ is a principal divisor on $X$.

In the previous two propositions, we did not prescribe any lifts for the points in the support of $D_3$ or $D_1$. However, in the lifting theorem allows us to prescribe all but $g$ lifts. In the following proposition we will do 