Tree Schreier generators are freely independent

Let $T$ be the rooted spanning tree corresponding to the given Schreier system by [L1]. If the positive edge from $Ht$ to $Htx$ lay in $T$, then the unique tree path from $H$ to $Htx$ would be the tree path to $Ht$ followed by that edge, so its label would be $tx$ and [L2] would give $s(t,x)=tx(tx{)}^{-1}=1$. Therefore every nontrivial Schreier generator corresponds to a unique positive edge outside $T$.

Take a nonempty reduced word in the nontrivial tree Schreier generators and their inverses. Replace each letter by its based loop from [L2] and concatenate those loops. Cancel adjacent inverse tree segments whenever they appear. Tree segments can disappear this way, but an edge outside $T$ can disappear only by meeting its own reverse immediately, which would mean that two consecutive Schreier generators were inverse letters, contrary to the reducedness of the word. So after all cancellations there remains a closed path whose label is a nonempty reduced word on $X\bigsqcup X^{-1}$.

By [L3], the nonempty reduced word from step 2.1 is not the identity in the ambient free group. Hence the original reduced word in the tree Schreier generators is nontrivial in $H$. Therefore those generators are freely independent.