Path as Parameterization of Contour/Corollary 1

Theorem
If $\gamma$ is a closed path, then $C$ is a closed contour.

Proof
By definition of closed path, we have


 * $\map \gamma a = \map {\gamma_1} {a_0} = \map {\gamma_n} {a_n}$

so:


 * $C_1$ has start point $\map \gamma a$

and:


 * $C_n$ has end point $\map \gamma a$.

By definition, it follows that $C$ is a closed contour.