Proof:

Let $\bar{e}$ denote the homotopy class of the constant loop based at $e.$ We first show that the operation $\circ$ is well defined. If $\gamma_1^\prime \sim \gamma_1^{\prime \prime}$ and $\gamma_2^\prime \sim \gamma_2^{\prime \prime}$ via the respective homotopies $F, G : I \times I \longrightarrow X$, it is easily checked that the map $F \cdot G : [0, 1]\times [0, 1]\longrightarrow X$ given by

$$F\cdot G (s, t) = F(s, t)\cdot G(s, t),$$

the product on the right denoting with group multiplication in $X$, is a homotopy between $\gamma_1^\prime(t) \gamma_2^\prime(t)$ and $\gamma_1^{\prime \prime}(t) \gamma_2^{\prime \prime}(t)$. We conclude that $\circ$ is a well defined binary operation on $\pi_1(X,e)$ with a two sided unit ${\bar e}$. Clearly, $\bar{e}$ is a common two sided unit element for both binary operations on $\pi_1(X,e)$. To invoke the lemma we show that the two binary operations are mutually distributive. Let $\gamma_1^\prime, \gamma_2^\prime \gamma_1^{\prime \prime}, \gamma_2^{\prime \prime}$ be loops based at $e$

$$(\;[\gamma_1^\prime][ \gamma_1^{\prime \prime}]\;) \circ (\; [\ga... ...\prime \prime})(t) \cdot (\gamma_2^\prime \ast \gamma_2^{\prime \prime})(t)\;]$$

We first verify through direct calculation that $(\gamma_1^\prime \ast \gamma_1^{\prime \prime}) \cdot (\gamma_2^\prime \ast \g... ...\gamma_2^\prime) \ast (\gamma_1^{\prime \prime} \cdot \gamma_2^{\prime \prime})$. Well,

$$(\gamma_1^\prime \ast \gamma_1^{\prime \prime})(t) \cdot (\gamma_2^\prime \ast \gamma_2^{\prime \prime})(t) = \gamma_1^{\prime}(2t) \gamma_2^{\prime}(2t), \phantom{XXXX} 0\leq t \leq \frac{1}{2}$$

$$= \gamma_1^{\prime \prime}(2t-1)\gamma_2^{\prime \prime}(2t-1), \frac{1}{2} \leq t \leq 1.$$

$$\therefore \quad [(\gamma_1^\prime ... ...^{\prime \prime})(t) \cdot (\gamma_2^\prime \ast \gamma_2^{\prime \prime})(t) ] = [\gamma_1^\prime(t) \cdot \gamma_2^\prime(t)] [\gamma_1^{\prime \prime}(t) \cdot \gamma_2^{\prime \prime}(t)]$$

So finally

$$([\gamma_1^\prime][\gamma_1^{\prime \prime}]) \circ ([\gamma_2^\p... ...\gamma_2^\prime])([\gamma_1^{\prime \prime}] \circ [\gamma_2^{\prime \prime}])$$

Thus lemma (12.1) is applicable for the binary operations $*$ and $\circ$ and the proof is complete.