Proposition 2

The symmetry $x \to x^{-1}$ is an isomorphism of the right uniformity onto the left uniformity.

This is an immediate consequence of Definition 1.

The reader should beware of supposing that the mapping $(x, y) \to xy$ of the uniform space $G_d \times G_d$ into the uniform space $G_d$ is in general uniformly continuous. Similarly, the symmetry $x \to x^{-1}$, considered as a mapping of $G_d$ onto $G_d$, is not in general uniformly continuous (see Exercises 3 and 4).