25. (Euclidean Algorithm in $\boldsymbol{Z}[\sqrt{-1}]$) Under the notation of the 24th problem, prove: if $\alpha_{1} \nmid \alpha_{0}$, then there must exist a positive integer $k$, such that the following division equations are obtained:
$$\begin{array}{l} \alpha_{0}=\eta_{1} \alpha_{1}+\alpha_{2}, \quad 0<N\left(\alpha_{2}\right)<N\left(\alpha_{1}\right), \eta_{1}, \alpha_{2} \in \boldsymbol{Z}[\sqrt{-1}], \\ \alpha_{1}=\eta_{2} \alpha_{2}+\alpha_{3}, \quad 0<N\left(\alpha_{3}\right)<N\left(\alpha_{2}\right), \eta_{2}, \alpha_{3} \in \boldsymbol{Z}[\sqrt{-1}], \\ \text{.................. } \\ \beta_{k-1}=\eta_{k} \alpha_{k}+\alpha_{k+1}, \quad 0<N\left(\alpha_{k+1}\right)<N\left(\alpha_{k}\right), \eta_{k}, \alpha_{k+1} \in \boldsymbol{Z}[\sqrt{-1}], \\ \beta_{k}=\eta_{k+1} \alpha_{k} \quad \eta_{k+1} \in \boldsymbol{Z}[\sqrt{-1}] . \end{array}$$