Proof:

Let $\gamma_1$ and $\gamma_2$ be two homotopic loops based at $x_0$ and let $F:I\times I\longrightarrow X$ be the homotopy fixing the base point $x_0$. Then $\sigma_i = F\circ T_i$ ($i = 1, 2$) are two singular two simplicies. It is an exercise to compute the boundary of these two singular simplicies and we find easily

$$\partial (\sigma_1) = \partial (F\circ T_1) = \varepsilon_{x_0} + \gamma_1 - F(t, t)$$

$$\partial (\sigma_2) = \partial (F\circ T_2) = \varepsilon_{x_0} + \gamma_2 - F(t, t).$$

The one chain $\gamma_1-\gamma_2$ is the boundary of the two chain $\sigma_1 - \sigma_2$ whence $\overline{\gamma}_1 = \overline{\gamma}_2$.