Corollary 2

Let B be a connected and locally connected topological space, let $(E, p)$ be a principal covering of B of group G. Let $E_0$ be a connected component of E and let $G_0$ be the subgroup of G stabilizer of $E_0$. The B-space $(E_0, p|E_0)$ is a principal covering of group $G_0($I, p. 103, prop. 6).

Every covering X of B which is associable to the principal covering E is associable to the principal covering $E_0$. In particular, the covering E is associable to the principal covering $E_0$.

Indeed, if the covering X is associable to E, the covering $p^*(X)$ is trivializable and the covering $p^*_0(X)$ induced by $p^*(X)$ above $E_0$ is therefore trivializable.

More precisely, note that the mapping $(x, g)\mapsto x\cdot g$ of $E_0\times G$ into E induces, by passing to the quotient, a B-isomorphism of principal coverings of $E_0\times^{G_0}G$ onto E.