I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. 
Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may take the form $[\mathbf{p}_0, \mathbf{p}_1,\dots, \mathbf{p}_k]$.
Can anyone explain this important moment?
I can't understand it by myself.