Let $n\in\mathbb N$. Let $x$ and $y$ be distinct indeterminates. Let $R$ be an integral domain. Let $\big(f_1(x),\dots,f_n(x)\big)\in R[x]^n$ be such that the ideal in $R[x]$ generated by $f_1(x),\dots,f_n(x)$ is $R[x]$. Also assume that $f_i(x)$ is monic for some $i\in\{1,\dots,n\}$.
If $g_1(x,y),\dots,g_n(x,y),h_1(x,y),\dots,h_n(x,y)\in R[x,y]$ and there exists an invertible $n\times n$ matrix $M$ with entries in $R[x,y]$ such that
$\begin{gather} \begin{bmatrix} g_1\\ \dots \\ g_n \end{bmatrix} = M \begin{bmatrix} h_1\\ \dots \\ h_n \end{bmatrix} \end{gather}$
then we write $(g_1,\dots,g_n)\sim(h_1,\dots,h_n)$.
Let $J$ be the set $\{c\in R\mid \big(f_1(x+cy),\dots,f_n(x+cy)\big)\sim \big(f_1(x),\dots,f_n(x)\big)\}$. Prove that if $c,d\in J$, then $c+d\in J$. Lang says this can be proven "easily."
By assumption \begin{align} \big(f_1(x+cy),\dots,f_n(x+cy)\big)&\sim \big(f_1(x),\dots,f_n(x)\big)\\ \big(f_1(x+dy),\dots,f_n(x+dy)\big)&\sim \big(f_1(x),\dots,f_n(x)\big) \end{align} By substituting $x+cy$ at in $x$ in the second expression, we get $$\big(f_1((x+cy)+dy),\dots,f_n((x+cy)+dy)\big)\sim \big(f_1(x+cy),\dots,f_n(x+cy)\big)$$ Consequently, \begin{align} \big(f_1(x+(c+d)y),\dots,f_n(x+(c+d)y)\big) &=\big(f_1((x+cy)+dy),\dots,f_n((x+cy)+dy)\big)\\ &\sim \big(f_1(x+cy),\dots,f_n(x+cy)\big)\\ &\sim \big(f_1(x),\dots,f_n(x)\big) \end{align} thus proving the assertion.