In a course I am currently taking, my professor performed the following steps in a derivation for the reachable canonical form of a transfer function.
Given $$H(s)=\frac{Y(s)}{U(s)}=\frac{b_1s^2+b_2s+b_3}{s^3+a_1s^2+a_2s+a_3}$$
$$H(s)=\frac{Y(s)}{U(s)}=\frac{b_1s^{-1}+b_2s^{-2}+b_3s^{-3}}{1+a_1s^{-1}+a_2s^{-2}+a_3s^{-3}}\frac{X(s)}{X(s)}$$
All good so far. But then my professor did this:
$$Y(s)=(b_1s^{-1}+b_2s^{-2}+b_3s^{-3})X(s)$$ $$U(s)=(1+a_1s^{-1}+a_2s^{-2}+a_3s^{-3})X(s)$$
I am highly suspicious that this is a legal operation, but I can't think of an easy counter example. Is this step ok?