A result I need is:
If $f(x,y) - f(x,z) = g(y,z)$ for all $(x,y,z)$, then $f(x,y) = a(x) + b(y)$ for some functions $(a,b)$.
This seems almost obvious, and I've constructed a proof, but that proof seems unnecessarily complicated and is remarkably tedious (and so isn't included here). I'd like pointers or ideas leading to something more simple and elegant. Surely this is a known result, or a special case of a well-known result?