Are positive real numbers $x,y$ allowed to be taken out during this proof?

Prove $$\left(x^2 – y^2\right)\left(\frac1y – \frac1x\right) \ge 0$$ where $x$ and $y$ are positive real numbers.

Can I simplify $\frac{(x^2 – y^2)(x-y)}{xy} \ge 0$
and then $(x^2 – y^2)(x-y) \ge 0$ cancelling out the $xy$?

Is this valid because then
\begin{align}
(x+y)(x-y)(x-y)&\ge 0\cdot(x-y)^2\ge 0
\end{align}
which is always true and then prove backwards from here?

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