In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text):
Let $\{a_1, \dots,a_n\}$ be a set of points in a compact Riemann surface $X$ and let $\{f_1,\dots,f_n\}$ be a set of principles parts (i.e., Laurent series with no positive exponents). Then the following are equivalent:
There exists a meromorphic function $f$ on $X$ such that the principle part of $f$ around $a_i$ is $f_i$, and $f$ has no additional poles.
$\sum_{i=1}^n \text{Res}(f_i\omega,a_i)=0$ for every holomorphic $1-$form $\omega$ on $X$
In the text, the proof of $2 \to 1$ is not mentioned, but it is stated that it "follows from the Serre Duality". Yet the Serre Duality follows from Riemann-Roch theorem, which relies on this proposition!
Is there a proof of this proposition which does not rely on Serre Duality or on Riemann-Roch Theorem?
This is V. Talovikova's text: https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Talovikova.pdf
