This theorem is used in the proof of our theorem.
Here is our theorem and its proof :
My question is that, how are the representations of $(118)$ and $(121)$ equal of each other?
Any help would be appreciated.
In (118), the summation is over all possible subsets of $\{1,\dots,p\}$, where I assume $dx_I$ is taken to mean $dx_{i_1} \wedge \dots \wedge dx_{i_k}$ and $\{i_1,\dots,i_k\}=I $.
In (121), $\alpha$ contains the subsets without $dx_p$, and the new summation contains all those with $dx_p$.