Path lifting in principal bundles

Question

Let $\pi : P \rightarrow M$ be a principal $G$-bundle and let $\gamma : [0,1] \rightarrow M$ be a smooth path. I define a lifting of $\gamma$ on $P$ in the following way: if $\{U_\alpha\}_\alpha$ is a trivializing open covering of $M$ I can find a partition $\{t_i\}_{i = 0}^k$ of $[0,1]$ in such a way that $\gamma([t_{i - 1},t_i]) \subset U_{\alpha_i}$ for each $i$. Now I can define the lifting of $\gamma$ as $\tau(s) := (\sigma_{\alpha_i} \circ \gamma)(s) \cdot g_i$ for $s \in [t_{i-1},t_i]$ where $\sigma_{\alpha_i} : U_{\alpha_i} \rightarrow P$ is a smooth local section and $g_i \in G$ is used to adjust the starting point of each segment. Now I want to prove that the path $\tau : [0,1] \rightarrow P$ is smooth but I am not sure how to prove the smoothness at the points $\{t_i\}_{i = 1}^{k - 1}$ (at the endpoints is obviously smooth).