ALet f$1,$ f$2,$ f$3$ map the set $\{$x$,$ y$\}$ to the set $\{1,1\}.$ Define A$1=$ f$1($x$)$f$2($x$)$f$3($x$),$ Prove that either A$1=1$ or one of A$2,$A$3,$A$4$ equals $+1.$