= Solution
Normalize the loop by
$$
u(t)=\frac{\gamma(t)}{|\gamma(t)|}\in S^1.
$$
Choose a continuous lift $\vartheta:[0,1]\to\mathbb R$ satisfying
$$
u(t)=e^{2\pi i\vartheta(t)}.
$$
Because $\gamma(0)=\gamma(1)$, the difference $\vartheta(1)-\vartheta(0)$ is an integer independent of the chosen lift. The <winding number of a continuous closed path> is
$$
\boxed{w(\gamma;0)=\vartheta(1)-\vartheta(0)}.
$$
If $\eta$ is a lift of $\delta/|\delta|$, then $\vartheta+\eta$ is a lift of
$$
\frac{\gamma\delta}{|\gamma\delta|}.
$$
Therefore
$$
\boxed{w(\gamma\delta;0)
=w(\gamma;0)+w(\delta;0)}.
$$
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