Let $\theta(G+H)$ denote the Lovász theta function of the vertex disjoint union of graphs $G$ and $H$. Prove that $\theta(G+H)\leq \theta(G) + \theta(H)$.
I believe that the stronger claim that $\theta(G+H)= \theta(G) + \theta(H)$ is also true as proven by Donald Knuth in section 18 of "the sandwich theorem".
However, I was wondering if there is a simpler way of showing that the weaker inequality above holds.