then the result follows from Lemma \[sec3lem3\].b; if $i\leq j$, then the result follows from Lemma \[sec3lem3\].a and \[sec3lem3\].c and the condition $ww'\neq 1$. Moreover, the S-polynomial $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ has a Gröbner representation in terms of $G$ if $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ have Gröbner representations in terms of $G$. Theorem \[sec2the11\] ensures that $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ has a Gröbner representation in terms of $G$. Note that the obstruction ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ can be either a multiple of a non-trivial obstruction or an obstruction without overlap (for instance, see Example \[sec3exa4\]). If ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ is a multiple of a non-trivial obstruction, then Lemma \[sec2lem6\] and Theorem \[sec2the11\] guarantee that $S_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ has a Gröbner representation in terms of $G$. If ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ is an obstruction without overlap, then, by Lemma \[sec2lem8\], its S-polynomial has a Gröbner representation in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\].

\[ncLWCrit\] Suppose that ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ are two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ such that there exist two words $w,w'$ in $\langle X\rangle$ satisfying $u_{s}=wv_{s}$ and $u'_{s}=v'_{s}w'$. Then ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ can be removed from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure if one of the following conditions 