6)[$\theta_{i_3}$]{} (5431,-1786)[$\theta_{i_2}$]{} (3226,-1786)[$1$]{} (6578,-879)[$\theta_{i_4}$]{} (5829,-459)[$\theta_{i_5}$]{} (5063,-1344)[$\theta_{i_1}$]{} (2003,-1801)[$X$]{} (2483,-616)[$Y$]{} (1042,-879)[$Z$]{} (601,-1561)[( 1, 0)[3300]{}]{}

Then by the argument above, we see that $\Psi_p((\theta_1, \ldots, \theta_6)) =
(P, Q, R)$. Since $w$ is arbitrary in ${{\bold H}}^2$, we get the conclusion.

Consider now the map $\Psi_{\langle 123456 \rangle} \times \Psi_{\langle 214356\rangle}:\Theta_6 \to
{\cal H} \times  {\cal H}$ which sends $\theta$ to $(\Delta_{\langle 123456
\rangle,\theta}$, $\Delta_{\langle 214356 \rangle,\theta})$. Then these two hexahedra can be seen in Fig. \[Fig:7\] stand on the first and third quadrants of the $uv$-plane. Notice that they have the common edge $(12)(34)56$ along the $w$-axis. Then we claim the following.

\[Lem:6injective\] The map $\Psi_{\langle 123456 \rangle} \times \Psi_{\langle 214356 \rangle}$ is injective.

Let $((P_1,Q_1,R_1),(P_2,Q_2,R_2))$ be an element in the image of $\Psi_{\langle 123456 \rangle} \times 
\Psi_{\langle 214356 \rangle}$ and $\theta
= (\theta_1,\ldots,\theta_6)$ any element in the preimage of $((P_1,Q_1,R_1),(P_2,Q_2,R_2))$. Set $w_1$ and $w_2$ the points in $\xi_{\langle 123456 \rangle}^{-1}((P_1,Q_1,R_1))$ and $\xi_{\langle 214356 \rangle}^{-1}((P_2,Q_2,R_2))$ corresponding to $\theta$ under the identification in Lemma \[Lem:6fibration\] respectively. Again, we see that the triangles $\triangle w_101$ and $\triangle w_201$ are congruent so that $w_1$ and $w_2$ are identical which we denote by $w$.

Let $X_i=P_i^2 $, $Y_i=1+ (w-1)Q_i^2 \in {{\bold H}}^2$ for $i=1,2$. Then the triangles $\triangle 0X_1w$ and $\triangle 0wX_2$ are similar since both have the external angles $\theta_{i_1}+\theta_{i_2}$, $\pi-\theta_{i_2}$, $\pi-\theta_{i_1}$. Similarly, the triangles $\triangle 10Y_1$ and $\triangle 1Y_20$ are similar since both of which have the external angles $\theta_{i_3}+\theta_{i_4}$, $\pi-\theta_{i_3}$, $\pi-\theta_{i_4}$. (see Fig.�