Proposition 6

Let $f : X \to Y$ be a proper mapping, and let K be a quasi-compact subset of Y. Then $\overline{f}^{-1}(K)$ is quasi-compact.

By Proposition 3 of no. 1, the mapping $f_K : \overline{f}^{-1}(K) \to K$ is proper. Since $K \to P$ is a proper mapping (Theorem 1, Corollary 1) it follows from no. 1, Proposition 5 a) that the composition $\overline{f}^{-1}(K) \xrightarrow{f_K} K \to P$ is proper, whence $\overline{f}^{-1}(K)$ is quasi-compact by Theorem 1, Corollary 1.