Theorem 7.6:

Suppose $\gamma_1,\; \gamma_2,\; \gamma_3$ are three paths in $X$ such that $\gamma_1(1)=\gamma_2(0);\; \gamma_2(1)=\gamma_3(0)$ then

$$(\gamma_1 \ast \gamma_2) \ast \gamma_3 \sim \gamma_1 \ast (\gamma_2 \ast \gamma_3)$$

Hence

$$([\gamma_1][\gamma_2])[\gamma_3]=[\gamma_1]([\gamma_2][\gamma_3])$$