I need help to prove the following function is positive for all $0<x<1$, $0<y<1$, $x\neq y$ and an integer $a\ge 2$,
$$f(x,y,a)=(y-1) x^a \left((a-2) x^2-(a-1) x (y+1)+a y\right)+(x-1)^2 x y^a.$$
According to all numerical testing of $f$ this claim is definitely true, but how can I formally prove it?