Let $$x+y=p$$ $$xy=q$$
Easy to proof that any fraction $F(p,q)=Z(p,q)/R(p,q)$, where $Z(p,q)$ and $R(p,q)$ are polynomials of $p$ and $q$ is symmetric with respect to $x$, $y$.
But let's say I can express $x$ and $y$ through $p$ and $q$ $$x = F(p,q) $$ $$y= f(p,q)$$
So we can express $$x-y=F-f$$
as we can see, the right part is symmetrical, but the left is antisymmetrical.
How is this possible?
For example $$2+3=5=p$$ $$2*3=6=q$$
$3=\frac{q}{2(q-p)}$
$2=\frac{q}{3(q-p)}$
$3-2=\frac{q}{6(q-p)}=\frac{q}{q}$