Solution
= Solution
Choose continuous argument lifts $\theta_1,\theta_2$ for the two curves. Their sum is a lift for their pointwise product, because
$$
\gamma_1(t)\gamma_2(t)=|\gamma_1(t)||\gamma_2(t)|e^{i(\theta_1(t)+\theta_2(t))}.
$$
Subtract its initial value from its final value to obtain
$$
\boxed{w(\gamma_1\gamma_2)=w(\gamma_1)+w(\gamma_2).}
$$
In particular, <winding numbers> subtract under pointwise division of curves avoiding zero.