A question about Nielsen–Schreier theorem I have a question about this well-known theorem about free groups by focusing on the proof stated by D. L. Johnson in his book "Presentations of groups ":
Theorem (Nielsen-Schreier): Let $F$ be a free group and $H$ a subgroup of $F$. Then $H $ is free.
In this book, he started the proof by:
Let $X$ be a set of free generators for $F$ and $U$ a Schreier transversal for $H$ (Lemma 
2). The resulting set $A$ generates $H$ (Lemma 3), and thus, so does the subset $B$ (Lemma 4). 

May I ask why "so does the subset $B$ "?



*

*Lemma 2. Every subgroup $H$ of $F$ has a Schreier transversal.

*Lemma 3. The elements of the set $A := \{ ux \overline{\mathbf{ux}}^{-1}  \mid u \text{ is in } U \text{ and } x \text{ in }X^{+}\cup X^{-} \}$ generate $H$.

*Lemma 4. We have $B := \{ ux \overline{\mathbf{ux}}^{-1} \mid u \text{ is in } U, x \text{ in } X \text{ and } ux \text{ does not belong to } U \}$ and $A \setminus \{e \}= B\cup B^{-1}$.
 A: I think these ideas are clarified by the use of groupoids, as giving more geometry to the algebra. See 
Higgins, P.J. Notes on categories and groupoids, Mathematical
  Studies, Volume 32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of   Categories, No. 7 (2005) pp 1--195.
ans also Topology and Groupoids. 
A Schreier transversal is there seen as a maximal tree in a covering graph. This method also uses the important idea of a covering  morphism of groupoids, which is equivalent  to the notion of action of a  groupoid on sets, and, of course, via the fundamental groupoid functor $\pi_1$ to the usual theory of covering maps of spaces, for sufficiently nice spaces. Thus, as is standard in algebraic topology, we use if possible an algebraic model of a topological situation. 
The argument goes as follows: let $F$ be a free group on a set of generators $S$. Let $H$ be a subgroup of $F$. The action of $F$ on the set $X$ of, say,  right cosets of $H$ in $F$ determines a covering morphism of groupoids $p: G \to F$, where the object set of $G$ is the set of right cosets. In particular, $p$ maps  the object group of $G$ at the object $H$, i.e. the coset $H1$, isomorphically to $H$. The inverse image by $p$ of the set $S$ of generators of $F$ is a graph $C(S)$. One then needs to show that $G$ is the free groupoid on the graph $C(S)$. This argument is an algebraic model of the usual  topological proof using covering spaces. 
This expansion to groupoids of the traditional realm of discourse has the advantage of allowing other notions, such as fibrations of groupoids, orbit groupoids, pushouts of groupoids, $\ldots$, which all have geometric,  topological and group theoretic applications. 
A: Every reduced word $w=a_1a_2\cdots a_t$ can be rewrote as
$$w=(\overline{e}a_1\overline{ea_1}^{-1})(\overline{a_1}a_2\overline{a_1a_2}^{-1})(\overline{a_1a_2}a_3\overline{a_1a_2a_3}^{-1})\cdots (\overline{a_1a_2\cdots a_{t-1}}a_t\overline{a_1a_2\cdots a_t}^{-1})$$
