Corollary 2

Let B be a topological space and let E be a covering of B; let us denote by $p$ its projection. Let $x$ be a point of E, and put $b=p(x)$. In order that the class in $\pi_1(B, b)$ of a loop $c$ in B at $a$ should belong to the image subgroup of the homomorphism $\pi_1(p, x)$, it is necessary and sufficient that the path $c'$ with origin $x$ lifting $c$ should be a loop of E at $x$.

The condition is obviously sufficient. Conversely, let $c'_1$ be a loop of E at $x$. Put $c_1=p\circ c'_1$ and suppose that the loops $c$ and $c_1$ are strictly homotopic. By Prop. 4, the path $c'$ has the same end point as the path $c'_1$. It is therefore a loop at $x$.