Lemma 9.1.4.label Let $G$ be a group and $\fU \subset 2^{G \times G}$ be a left/right translation-invariant uniformity, then $\fU$ admits a fundamental system of symmetric, left/right translation-invariant entourages.
Proof. Let $z \in G$, then the map $(x, y) \mapsto (zx, zy)$ is a bijection. Thus for any translation-invariant entourages $U, V \in \fU$, $(U \cap V) = zU \cap zV$, and $U \cap V$ is left translation-invariant. By Lemma 6.1.9, $\fU$ admits a left fundamental system of symmetric, translation-invariant entourages.$\square$
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