For (i): since $f$ is nonzero on $\Gamma$ and $\Gamma$ is a closed curve, the angle $\psi$ of $f$ along $\Gamma$ must return to its initial value modulo $2\pi$ after one traversal. The net change is therefore an integer multiple of $2\pi$, making $I_\Gamma$ an integer. For (ii): if $f$ is nonzero on the interior of $\Gamma$ and on $\Gamma$ itself, then $\psi$ extends to a continuous function on the interior. By the invariance property (iii), $\Gamma$ can be contracted to a point without crossing any zero, during which the index varies continuously among integers; it is therefore constant. As $\Gamma$ shrinks to a point, the change in $\psi$ along $\Gamma$ tends to zero, forcing $I_\Gamma = 0$. For (iii): a continuous deformation of $\Gamma$ that avoids fixed points gives a homotopy of the map $\Gamma \to S^1$ defined by $x \mapsto f(x) / |f(x)|$; the degree of this map is $I_\Gamma$ and is a homotopy invariant. For (iv): reversing the traversal direction changes the sign of $d\psi$ at every point and the sign of the integration measure, leaving the integral unchanged.