Lemma 3.2.3.8. Let $\alpha $, $\beta $, and $\gamma $ be elements of $\pi _{n}(X,x)$. For $2 \leq i \leq n$, we have $m_ i( \alpha , m_{i-1}(\beta , \gamma ) ) = m_{i-1}( \beta , m_ i(\alpha , \gamma ) )$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Applying Lemma 3.2.3.7 to the tuple $( [e], \ldots , [e], \beta , m_{i-1}(\beta , \gamma ), \gamma , [e], \ldots , [e] )$, we deduce that the tuple $( [e], \ldots , [e], \beta , m_ i( \alpha , m_{i-1}(\beta , \gamma ) ), m_ i( \alpha , \gamma ), [e], \ldots , [e] )$ bounds, which is equivalent to the asserted identity. $\square$