Corollary 2

Tag 07AA, permanent. Cite it as [Bourbaki, Tag 07AA].

Topological Vector Spaces, chapter I, § 3. Metrisable topological vector spaces

Let E be a vector space over a non-discrete valued division ring, let $\mathscr{T}_1$ and $\mathscr{T}_2$ be two topologies on E compatible with its vector space structure and for each of which E is metrisable and complete. Then, if $\mathscr{T}_1$ and $\mathscr{T}_2$ are comparable, they are identical.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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