I am trying to understand the assertions made at the end of the paragraph below. I am not sure if I am missing some certain facts from previous chapters, but I do not see the claims made in bold typeface. Could you please guide me?
Let $K/k$ be a finite algebraic extension and let $|\hspace{0.2cm}|$ be a valuation on $k.$ ... We do not suppose that $k$ is complete and ask ourselves what extensions, if any, there are of $|\hspace{0.2cm}|$ to $K$ ... Suppose that the valuation $|\hspace{0.2cm}|_{1}$ on $K$ extends $|\hspace{0.2cm}|$ and let $K_1$ be the completion of $K$ with respect to it. Then $K_1$ contains the completion $\bar{k}$ of $k$ with respect to $|\hspace{0.2cm}|.$ A basis $\{B_i\}$ of $K/k$ clearly generates $K_1$ as a $\bar{k}$-vector space.