What is the Mathematical Property that justifies equating coefficients while solving partial fractions? The McGraw Hill PreCaculus Textbook gives several good examples of solving partial fractions, and they justify all but one step with established mathematical properties.
In the 4th step of Example 1, when going from:
$$1x + 13 = (A+B)x+(4A-5B)$$
they say to "equate the coefficients", writing the linear system
$$A+B = 1$$
$$4A-5B=13$$
It is a simple step, color coded in the textbook for easy understanding, but McGraw Hill does not justify it with any mathematical property, postulate or theorem.  Addition and/or multiplication properties of equality don't seem to apply directly.
Can someone help me justify this step?!

Lemma: If $px+q=0$ for all values of $x$, then $p=q=0$.
Proof: In particular, $p(0)+q=0$, which means that $q=0$. So $px=0$ for all $x$, which means that $p(1)=0$, and so $p=0$.
Theorem: If $px+q$ and $rx+s$ are equal for all values of $x$, then $p=r$ and $q=s$.
Proof: If $px+q$ and $rx+s$ are equal for all values of $x$, then
$$
px+q-(rx+s)=0
$$
for all values of $x$. But this expression can be rewritten as
$$
(p-r)x+(q-s)
$$
and so, by the lemma, $p-r=0$ and $q-s=0$. That is, $p=r$ and $q=s$.
