Let $U\subseteq \mathbb{C}$ open, and $f:U\to \mathbb{C}$ holomorphic. Let $\varphi :R\to U$ be a $C^1$ image of a rectangle. Then given any parametrization of $\varphi (\delta R)$, say $\gamma$, we have:
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PROOF:

1. See steps for 11. Lemma 2.1.1- Cauchy Integral Theorem for rectangles.
2. the only tricky part is that we need to bound the diameter of $\varphi (R)$ and the length. We can do this since $\varphi$ is continuously differentiable, and so the operator norm of the Jacobian has a maximum value on $R$, say $C$. We use this constant to bound $diam(\varphi (R^{(n)}))\le C\frac{diam(R)}{2^n}$