proof of closed curve theorem

Let

$$f(x+iy)=u(x,y)+iv(x,y).$$

Hence we have

$${\int }_{C}f(z)𝑑z={\int }_C\omega +i{\int }_C\eta$$

where $\omega$ and $\eta$ are the differential forms

$$\omega =u(x,y)dx-v(x,y)dy,\eta =v(x,y)dx+u(x,y)dy.$$

Notice that by Cauchy-Riemann equations $\omega$ and $\eta$ are closed differential forms. Hence by the lemma on closed differential forms on a simply connected domain we get

$${\int }_{C_1}\omega ={\int }_{C_2}\omega ,{\int }_{C_1}\eta ={\int }_{C_2}\eta .$$

and hence

$${\int }_{C_1}f(z)𝑑z={\int }_{C_2}f(z)𝑑z$$