I have the following situation: Let $L_1,L_2,L_3$ and $L_4$ be linear forms in $\mathbb{C}^k$ such that $\frac{L_1L_2}{L_3}=L_4$. Can we conclude from here that $L_3$ is a multiple of $L_1$ or of $L_2$? This is equivalent to ask if the factorization of $L_1L_2$ is unique.
I read in a commentary of another question that this is true but there was no proof. Thanks for the help!