Lemma 3

Let G be a groupoid and let $p, q$ be integers $\geqslant 1$. For every pair $(j, k)$ of integers such that $0\leqslant j\leqslant p$ and $0\leqslant k\leqslant q$, let $x_{j,k}$ be a vertex of G; for every pair $(j, k)$ such that $1\leqslant j\leqslant p$ and $0\leqslant k\leqslant q$, let $h_{j,k}$ be an arrow of G joining $x_{j-1,k}$ to $x_{j,k}$; for every pair $(j, k)$ such that $0\leqslant j\leqslant p$ and $1\leqslant k\leqslant q$, let $v_{j,k}$ be an arrow of G joining $x_{j,k-1}$ to $x_{j,k}$.

Assume that, for every pair $(j, k)$ of integers such that $1\leqslant j\leqslant p$ and $1\leqslant k\leqslant q$, the arrows $v_{j-1,k}$ and $h_{j,k}$ are composable, analogously the arrows $h_{j,k-1}$ and $v_{j,k}$, and that one has $v_{j-1,k}h_{j,k}=h_{j,k-1}v_{j,k}$. Then,

Let us first treat the particular case where $q= 1$ and prove the result by induction on $p$. If $p= 1$, the assertion to be proved is true by assumption; suppose it true for $p-1$; one then has

by the induction hypothesis, whence the relation for $p$.

Let us now prove the result by induction on $q$. It is true for $q= 1$ by what precedes; if it is true for $q-1$, one then has

From the case $q= 1$, one has

whence the required relation.