 Solution

Normalize the loop byChoose a continuous lift $\vartheta :[0,1]\to \mathbb{R}$ satisfyingBecause $\gamma (0)=\gamma (1)$, the difference $\vartheta (1)-\vartheta (0)$ is an integer independent of the chosen lift. The winding number of a continuous closed path is

If $\eta$ is a lift of $\delta /|\delta |$, then $\vartheta +\eta$ is a lift ofTherefore