Holomorphic integrals agree on homologous cycles

Given: An open $\mathrm{\Omega }$, a holomorphic $f:\mathrm{\Omega }\to \mathbb{C}$, and cycles ${\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1$ with traces in $\mathrm{\Omega }$, homologous in $\mathrm{\Omega }$.

If $\mathrm{\Gamma }$ is a cycle with trace in an open $\mathrm{\Omega }$, null-homologous in $\mathrm{\Omega }$, and $f$ is holomorphic on $\mathrm{\Omega }$, then ${\int }_{\mathrm{\Gamma }}fdz=0$ (Cauchy's theorem for a null-homologous cycle).

Two cycles with traces in $\mathrm{\Omega }$ are homologous in $\mathrm{\Omega }$ when their difference is null-homologous in $\mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

$({\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2{)}^{\ast }={\mathrm{\Gamma }}_1^{\ast }\cup {\mathrm{\Gamma }}_2^{\ast }$ and $(-\mathrm{\Gamma }{)}^{\ast }={\mathrm{\Gamma }}^{\ast }$; the sum of two cycles and the negative of a cycle are cycles; and for $f$ continuous on the traces involved, ${\int }_{{\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2}fdz={\int }_{{\mathrm{\Gamma }}_1}fdz+{\int }_{{\mathrm{\Gamma }}_2}fdz$ and ${\int }_{-\mathrm{\Gamma }}fdz=-{\int }_{\mathrm{\Gamma }}fdz$ (Chain integration and the index are additive in the chain, and reverse with it).

${\int }_{\mathrm{\Gamma }}fdz={\sum }_{k<r,m_k\mathrm{\ne }0}{m}_k{\int }_{{\gamma }_k}fdz$ (Integration over a complex chain and the index of a chain), a chain being a finite list of integer-weighted complex contours (Complex chains, their traces, and cycles).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there).