Let \(z, z^{\prime} \in \mathbb{R}^{N}\) be two portfolios with riskless payoffs, i.e.,

\[\sum_{n=1}^{N} d_{s n} z_{n}=c \quad \text { and } \quad \sum_{n=1}^{N} d_{s n} z_{n}^{\prime}=c^{\prime}\]

for some \(c, c^{\prime}>0\), for all \(s=, 1 \ldots, S\). Show that, if there are no arbitrage opportunities, then \(\frac{c}{p^{\top} z}=\frac{c^{\prime}}{p^{\top} z^{\prime}}\) (i.e., the two portfolios \(z\) and \(z^{\prime}\) have the same rate of return).