Path integral on Riemann surface is determined only by homology class of path

Question

I heard thanks to Cauchy theorem, we can say path integral on Riemann surface is determined only by homology class of the path.

That is, let $ω$ be holomorphic 1-form differential on Riemann surface $X$.∮ω on the path $r$ and $∮ω$ on the path $r'$ are the same if two paths $r$ and $r'$ are homologous.

I understand in this direction, we use cauchy theorem. But how to prove the other direction, that is, how to prove

if integral are the same, the paths are homologous.

Thank you in advance.