Let

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

Hence we have

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

where $\omega $ and $\eta $ are the differential forms^{}

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

Notice that by CauchyRiemann 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

$${\int}_{{C}_{1}}f(z)\mathit{d}z={\int}_{{C}_{2}}f(z)\mathit{d}z$$ 
