Proposition 10

The canonical bijection of the subspace $\varphi(A)$ of $X/G$ onto the orbit space $A'/G$ is a homeomorphism.

Now let $S$ be an equivalence relation on $X$ such that, for each $s \in G$, the mapping $x \to s.x$ is compatible with $S$ [in other words, such that the relation $x \equiv y \pmod{S}$ implies $s.x \equiv s.y \pmod{S}$]; for the sake of brevity we shall say that the relation $S$ is compatible with the group $G$. If $\psi$ is the canonical mapping of $X$ onto $X/S$, and if $s.\psi(x)$ denotes the class mod $S$ of $s.x$, then $X/S$ has $G$ as groups of operators with respect to the external law $(s, \psi(x)) \to s.\psi(x) = \psi(s.x)$. Moreover: