I am looking at the slides from this here. On slide 5 they have the following
$$ (-1)^{m+l} \sum_{k \leq l}\left(\begin{array}{c}{l-k} \\ {m}\end{array}\right)\left(\begin{array}{c}{s} \\ {k-n}\end{array}\right)(-1)^{k-m+l} $$
They then state the following on slide 6,
$$ \begin{array}{l}{\text { Now since } k \leq l, l, m, n \in \mathbf{Z} ; \text { and } l, m, n \geq 0 \text { we }} \\ {\text { can use our symmetry property on the left hand }} \\ {\text { side to obtain: }} \\ {(-1)^{l+m} \sum_{k \leq l} \left(\begin{array}{c}{l-k} \\ {l-k-m}\end{array}\right)\left(\begin{array}{c}{s} \\ {k-n}\end{array}\right)(-1)^{l-k-m}}\end{array} $$
My question is then, why are they rewriting the very last exponent from $k-m+l$ to $l-k-m$?. Maybe i am missing something fundamental butthis doesn't seem clear to me given their definition of the symmetry property in the beginning of the slides.