Additional information concerning the content of the product of polynomials

Let $\operatorname{Cont}_Af$ denote the content of the polynomial $f$ in several unknowns with coefficients from the extension $R$ of the ring $A$. We prove that for arbitrary polynomials $f$ and $g$ the relation
$$ \operatorname{Cont}\nolimits_Afg\cdot(\operatorname{Cont}g)^m=\operatorname{Cont}f\cdot(\operatorname{Cont}g)^{m+1}, $$
holds, where $m+1$ is the number of the nonzero terms of the polynomial $f$.