right coprime factorization

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The following is from a textbook:

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To check the example above, we have to fine $x,y \in RH^\infty$ such that $xm+yn=1$. (A definition of $m,n$ being right co-prime).

However, I have difficulty in finding it since to cancel the numerator of $n,m$, both $x,y$ cannot $\in RH^\infty$.

How to find such $x$ and $y$?