Schreier index formula

Let $F$ be a free group of finite rank, and let $H$ be a subgroup (http://planetmath.org/Subgroup) of finite index in $F$.By the Nielsen-Schreier theorem, $H$ is free.The Schreier index formula states that

$$\mathrm{rank}(H)=|F:H|\cdot (\mathrm{rank}(F)-1)+1.$$

This implies more generally that if $G$ is a group generated by $m$ elements, then any subgroup of index $n$ in $G$ can be generated by at most $nm-n+1$ elements.