Relationship between nabla exponential and delta exponential

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Theorem: If $p$ is continuous and $\nu$-regressive then $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ where $\hat{e}_p$ denotes the $\nabla$-exponential and $e_p$ denotes the $\Delta$-exponential.

Proof: