On products of Weierstrass sigma functions

We prove the following result. Let $f\colon\mathbb C\to\mathbb C$ be an even entire function. Let there exist $\alpha_j,\beta_j\colon\mathbb C\to\mathbb C$ with
$$ f(x+y) f(x-y) = \sum_{j=1}^4\alpha_j(x)\beta_j(y),\qquad x,y\in\mathbb C. $$
Then $f(z)=\sigma_L(z)\cdot\sigma_\Lambda(z)\cdot e^{Az^2+C}$, where $L$ and $\Lambda$ are lattices in $\mathbb C$, $\sigma_L$ is the Weierstrass sigma function associated to the lattice $L$, and $A,C\in\mathbb C$.