Statement: If a vector field R is irrotational then a line integral is independent of path.
Proof. Let $\nabla$ $\times$ $\vec A=0$ in $R$ consider the difference of two line integral from the point $r_0$ to $r$ along two curves $C_1$ & $C_2$.
$$\int_{C_1} \vec A.dr_1 -\int_{C_2} \vec A.dr_2$$
where $r_1$ is integration variable to distinguish it from the limit of integration $r$ &$r_0$
now $$\int_{C} \vec A.dr_1 = \int_{C_1} \vec A.dr_1 -\int_{C_2} \vec A.dr_2$$
from Stokes theorem $\int_{C} \vec A.dr_1$ =$\int_{S} \nabla \times \vec A.\vec ds $
$\nabla \times \vec A=0 $
so $\int_{C} \vec A.dr_1=0$
Is there any mistake?