Before revision: We provide elementary proofs of results from \mbox{%DIFAUXCMD Saari0pt%DIFAUXCMD and \mbox{%DIFAUXCMD DEMO }0pt%DIFAUXCMD regarding }  the possible outcomes arising from a fixed profile within the class of positional voting systems.
After revision: We provide elementary proofs of results from \mbox{%DIFAUXCMD Saari0pt%DIFAUXCMD and \mbox{%DIFAUXCMD DEMO }0pt%DIFAUXCMD regarding } several results concerning the possible outcomes arising from a fixed profile within the class of positional voting systems.