# Kerodon

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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

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