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 $A \setminus \{e \}= B\cup B^{-1}$.