English: Example of the topological proof of the Nielsen-Schreier Theorem. The free group G = pi_1(X) has two generators corresponding to loops a,b from the base point P in X. The index 2 subgroup H of even-length words corresponds to the covering graph Y with two vertices corresponding to the cosets H and H' = aH = bH = a^{-1}H = b^{-1}H, and two lifted edges for each of the original loop-edges a,b. Contracting one of the edges of Y gives a homotopy equivalence to a bouquet of 3 circles, so that H = pi_1(Y) is free on three generators.