The nontrivial Schreier generators generate the subgroup

Let $h\in H$, and choose any word $w$ on $X\bigsqcup X^{-1}$ representing $h$. By [L3], free-reducing $w$ does not change its rewrite, so we may assume $w=a_1\cdots a_n$ is reduced. If $t_j$ denotes the chosen representative of the coset of the prefix $a_1\cdots a_j$, and if ${\sigma }_j$ is the $j$th Schreier rewriting factor from [L1], then $t_{j-1}{a}_j={\sigma }_j{t}_j$ for every $j$.

Multiplying the identities from step 1.1 yields $h=w={\sigma }_1\cdots {\sigma }_n{t}_n=\tau (w)t_n$. Because $h\in H$, the last coset is $H$, so the final representative is $t_n=1$. Thus $h=\tau (w)$ is a product of Schreier generators and their inverses.

By [L2], each Schreier generator belongs to $H$, so the same is true for its inverse. After deleting the trivial factors in the product from step 2.1, we obtain an expression for $h$ as a product of nontrivial Schreier generators and their inverses. Therefore those nontrivial generators generate $H$.