 Solution

Fix $w\in \mathbb{C}$ and apply the stated clockwise argument principle to $h=f-w$. Its logarithmic derivative is periodic, so the integrals over opposite sides of a fundamental cell cancel. Hence $P-Z=0$. The poles of $f-w$ are precisely the poles of $f$, with unchanged multiplicities, while its zeros are the solutions of $f(z)=w$. Thus the value multiplicity of an elliptic function is the same fixed pole count for every finite $w$, counting multiplicity.