Alternating mapping/Swap property/Fact/Proof

Due to the definition of alternating and fact, we have

${\displaystyle {}{\begin{aligned}0&=\Phi (v_{1},\ldots ,v_{r-1},v_{r}+v_{s},v_{r+1},\ldots ,v_{s-1},v_{r}+v_{s},v_{s+1},\ldots ,v_{n})\\&=\Phi (v_{1},\ldots ,v_{r-1},v_{r},v_{r+1},\ldots ,v_{s-1},v_{s},v_{s+1},\ldots ,v_{n})+\Phi (v_{1},\ldots ,v_{r-1},v_{s},v_{r+1},\ldots ,v_{s-1},v_{r},v_{s+1},\ldots ,v_{n}).\end{aligned}}}$