Proposition 7

— Let $f \in A[X]$ be a non-zero polynomial in an indeterminate and $a$ its leading coefficient. If $a$ is cancellable in A (in particular iff is monic) then we have for any non-zero element'g of $A[X]$,

Let $g \in A[X]$ be a non-zero polynomial, $\beta$ its leading coefficient, $n = \deg f$ and $p = \deg g$. Then the coefficient of $X^{n+p}$ in $fg$ is $\alpha \beta$ which does not vanish, whence the proposition.