Corollary 3

Let X be a topological space locally connected by arcs, Y an unravelable topological space, and let $f: X\rightarrow$ Y be a continuous relatively connected mapping. For every point $x$ of X, the homomorphism $\pi_1(f, x):\pi_1(X, x)\rightarrow \pi_1(Y, f(x))$ is surjective.

By IV, p. 353, prop. 4, one may suppose that the spaces X and Y are connected. The corollary then follows from proposition 5 and proposition 2 of IV, p. 352.