Apparently the following holds: $$ \sum_{k=0}^{m-1} [x|k] \left(a_{m-k}(x) - a_{m-k-1}(x)\right) = 1-x[x|m] $$ where $a_k(x) = k[x>k]$ and $[...]$ is the conditional bracket i.e. $=1$ if it is true and $0$ otherwise.
How should one proceed in proofing it?