Proposition 5

Let B be a nonempty topological space and let E be a Galois covering of B. The mapping $(h, x)\mapsto h(x)$ of Aut$_B(E)\times E$ into E is a right operation law of the group Aut$_B(E)^{\circ}$ on E which makes E into a principal covering with group Aut$_B(E)^{\circ}$.

Let us endow the group Aut$_B(E)^{\circ}$ with the discrete topology; the operation law $(h, x)\mapsto h(x)$ is then continuous. This operation is free (I, p. 34, Corollary 2 of Proposition 11). Since E is a Galois covering of B, its fibres are the orbits of this operation. By Proposition 4 of I, p. 97, E is a principal covering with group Aut$_B(E)^{\circ}$.