Corollary 1

— If A is an integral domain, and $u, v$ are elements of $A[(X_i)_{i \in I}]$, then $\deg(uv) = \deg u + \deg v$.

We can limit ourselves to the case where $u$ and $v$ are non-zero. Let $m = \deg u, n = \deg v$; we have

where $u_h$ (resp. $v_h$) is the homogeneous component of degree $h$ of $u$ (resp. v). Since $u_m \neq 0$ and $v_n \neq 0$, we have $u_m v_n \neq 0$ (Prop. 8). Now $uv = u_m v_n + w$ with $\deg w < m + n$, whence the result.