About the expected value of a sum of random variables

Question

In the textbook I'm reading, it states $$E[g(X,Y)+h(X,Y)]=E[g(X,Y)]+E[h(X,Y)]$$ as a theorem, and derives $$E[X+Y]=E[X]+E[Y]$$ as a corollary by putting $g(X,Y)=X$ and $h(X,Y)=Y$. But isn't the opposite more natural? That is, if $$E[X+Y]=E[X]+E[Y]$$ holds for any random variables $X$ and $Y$, then since $g(X,Y)$ and $h(X,Y)$ are themselves random variables, $$E[g(X,Y)+h(X,Y)]=E[g(X,Y)]+E[h(X,Y)]$$ holds. Am not understanding something?