to be Garside and are endowed with a suitable length function.

This condition implies the extra condition on $\varphi$ in section \[sectembed\], namely that, if $n \in M(S)$ divides $\varphi(m)$ for some $m \in M(S_0)$, then $n \in \varphi(M(S_0))$. Indeed, if we have such $n,m$,then $n \in \varphi(M(S_0))$ unless $n$ can be written as a word containing some $x \in S \setminus S_0$. But in that case $\varphi(m)$ can also be written as a word in $S$ containing $x$. Now note that the defining relations involving such a $x$ cannot make it disappear, except when $x \in \mathcal{C}$. By contradiction this settles the cases $S_0 \cap \mathcal{C} = \emptyset$ and $S_0
\supset \mathcal{C}$. In case $S_0 \cap \mathcal{C} = \{ t_{i} \}$, we can assume $x = t_j$ for $j \neq i$, and would get equality in $M(S)$ of two words on $S$, one involving $t_i$ and no other element of $\mathcal{C}$, and the other involving $t_j$. But we check on the defining relations that all relations involving $t_i$ either involve only $t_i$ and no other elements of $\mathcal{C}$ in which case they preserve that property and do not make the $t_i$’s disappear, or they involve several elements of $\mathcal{C}$ 