Theorem 16.2 (Covering homotopy property):

Let $p: {\tilde X}\longrightarrow X$ be a covering projection and ${\tilde x}_0\in {\tilde X}, x_0 \in X$ be chosen base points such that $p({\tilde x}_0) = x_0$. Let $\gamma_1,\gamma_2$ be two curves in $X$ starting at $x_0$ and having the same terminal points and $F: [0, 1]\times [0, 1] \longrightarrow X$ be a homotopy between $\gamma_1$ and $\gamma_2$. There is a unique lift ${\tilde F}: [0, 1]\times [0, 1]\longrightarrow {\tilde X}$ of $F$ such that ${\tilde F}(0, 0) = {\tilde x}_0$. In particular the unique lifts of $\gamma_1$ and $\gamma_2$ starting at ${\tilde x}_0$ have the same terminal points.