Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Lemma 3.2.3.9. Let $\alpha $ and $\beta $ be elements of $\pi _{n}(X,x)$. For $2 \leq i \leq n$, we have $m_{i}(\alpha , \beta ) = m_{i-1}( \beta , \alpha )$.

Proof. Combining Lemma 3.2.3.8 with Example 3.2.3.6, we obtain

\[ m_ i( \alpha , \beta ) = m_ i( \alpha , m_{i-1}(\beta , [e] ) ) = m_{i-1}( \beta , m_ i( \alpha , [e] ) ) = m_{i-1}( \beta , \alpha ). \]
$\square$