Corollary 3

Let B be a topological space and let $(E_1, p_1)$ and $(E_2, p_2)$ be coverings of B. Let us denote by $(E, p)$ their fibre product over B, and let $x= (x_1, x_2)$ be a point of E. The image of the homomorphism $\pi_1(p, x)$ is the intersection of the images of the homomorphisms $\pi_1(p_1, x_1)$ and $\pi_1(p_2, x_2)$.

Let $c$ be a loop in B at $p_1(x_1) =p_2(x_2)$ and, for $i\in \{1,2\}$, let $c'_i$ be the path with origin $x_i$ in $E_i$ lifting $c$. The fibre product over B $E = E_1\times_BE_2$ is a covering of B (I, p. 72, cor. 3 of Prop. 2), and the path $c':t\mapsto$ $(c'_1(t), c'_2(t))$ is the unique path with origin $x$ in E which lifts $c$. In order that $c'$ should be a loop, it is necessary and sufficient that $c'_1$ and $c'_2$ should be so. Corollary 3 thus follows from Corollary 2.