just that, using the full power of Theorem \[lifting thm convenient\].

\[D2 lift\] Given lifts in $X(K)$ for every point in $\{a_e, b_e, v(e), w(e), p_e, q_e \mid e \in E \}$, which we denote by $a'_e, b'_e, v(e)', w(e)', p'_e, q'_e$ respectively, there exist for every $i = 1,\dots,g$ lifts $c'_{e_i}$ and $d'_{e_i}$ of $c_{e_i}$ and $d_{e_i}$ such that $$\begin{aligned}
D_2' \coloneqq \sum_{e \in E} s(e) \left( v(e)' + w(e)' - p_e' - q_e' \right) + 
\sum_{i = 1}^g -c_{e_i}' + a_{e_i}' + b_{e_i}' - d_{e_i}'\end{aligned}$$ is a principal divisor on $X$.

All three Propositions follow directly from Theorem \[lifting thm convenient\].

We let $f_1, f_2, f_3 \in K(X)$ be such that $\operatorname{div}(f_i) = D'_i$ so that $\log \vert f_i \vert|_{\Gamma} = D_i$. Let $U$ be the open set of $X$ obtained by removing all of the points $v'(e)$, $w'(e)$, $a_e'$, $b_e'$, $c_e'$, $d_e'$, $p_e'$, $q_e'$ for each edge $e$. Then we have the map $$\begin{aligned}
f \coloneqq (f_1, f_2, f_3) \colon U \to {\mathbf}G_m^3.\end{aligned}$$ For every three-dimensional, proper toric variety $Y$, this map extends to a morphism $$\begin{aligned}
\varphi \colon X \to Y.\end{aligned}$$

\[prop inj implies faithful\] Assume that for a vertex $v$ of $\Sigma(\varphi)$, the number of 