Lemma 4

Let G be a connected topological group, X a Hausdorff topological space and $f_1, f_2$ laws of left (resp. right) operation of G on X such that, for all $x \in X$, the mappings $s \mapsto f_1(s, x),\ s \mapsto f_2(s, x)$ of $G$ into $X$ are continuous. Suppose that there exists a neighbourhood $V$ of $\{e\} \times X$ in $G \times X$ such that $f_1$ and $f_2$ coincide on $V$. Then $f_1 = f_2$.

Let $x \in X$ and $A$ be the set of $g \in G$ such that $f_1(g, x) = f_2(g, x)$. Then $A$ is closed in $G$. On the other hand, let $g \in A$; we write $y = f_1(g, x) = f_2(g, x)$. There exists a neighbourhood $U$ of $e$ in $G$ such that $f_1(t, y) = f_2(t, y)$ for $t \in U$, in other words such that $f_1(t', x) = f_2(t', x)$ for $t' \in Ug$ (resp. $gU$). Hence $A$ is open in $G$ and therefore $A = G$.