Relationship between delta exponential and nabla exponential

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Theorem: Let $p \in \mathcal{R}$ and $t,s \in \mathbb{T}$, then the following formula holds: $$\left( \dfrac{1}{e_p(\cdot,s)} \right)^{\Delta} = -\dfrac{p(t)}{e_p^{\sigma}(\cdot,s)}.$$

Proof: