13. (Euclidean algorithm in $Q[x]$) Let $f_{0}, f_{1} \in Q[x]$, $f_{1} \neq 0, f_{1} \nmid f_{0}$. Prove:
(i) It is always possible to repeatedly apply the division algorithm from the previous problem to obtain the following $k+1$ equations:
$$\begin{array}{ll} f_{0}=q_{0} f_{1}+f_{2}, & \operatorname{deg} f_{2}<\operatorname{deg} f_{1}, f_{2} \neq 0, \\ f_{1}=q_{1} f_{2}+f_{3}, & \operatorname{deg} f_{3}<\operatorname{deg} f_{2}, f_{3} \neq 0, \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ f_{k-1}=q_{k-1} f_{k}+f_{k+1}, & \cdots \cdots \cdots \cdots \cdots \cdots \\ f_{k}=q_{k} f_{k+1} ; & \operatorname{deg} f_{k+1}<\operatorname{deg} f_{k}, f_{k+1} \neq 0, \end{array}$$
(ii) $f_{k+1} \mid f_{j}(0 \leqslant j \leqslant k)$;
(iii) For each $j(0 \leqslant j<k)$, there exist $h_{j}, h_{j+1} \in Q[x]$, such that
$$f_{k+1}=h_{j} f_{j}+h_{j+1} f_{j+1} .$$