Corollary 1

Under the hypotheses of Proposition 9, if $A$ is a subset of $G$ which is a complete subspace with respect to the uniformity $U_2$ corresponding to $\mathfrak{T}_2$, then $A$ is also a complete subspace with respect to the uniformity $U_1$ corresponding to $\mathfrak{T}_1$.

If $A_1$ is the closure of $A$ in $G_1$, then $f(A_1)$ is contained in the closure of $A$ in $G_2$, which by hypothesis is equal to $A$. Since $f(A) = A$ by definition and $f$ is injective, we have $A_1 = A$.