Let $D=(V,E)$ be some basic flow network and $f$ some flow. I’m trying to prove the following lemma:

I’ve been told that this is a proof that follows directly from the definitions of flow yet I can’t possibly how exactly that’s true; I may just be unknowingly misinterpreting some definitions. Anyway, in my attempt to prove this lemma I first noted that $text{Netflow(v)}=0$, for all $v$ that are not the sink or source. This would then imply that $sumlimits_{v in S} {text{Netflow(v)}}$ is equal to $0$, $text{Netflow(source)}$, $text{Netflow(sink)}$ or $text{Netflow(source)} + text{Netflow(sink)}$, depending on whether $S$ contains the source or the sinks, or both. However, I was not able to relate this in anyway to $$sumlimits_{e in E(S,S’)} f(e) – sumlimits_{e in E(S’,S)}{f(e)}$$. I think I’m missing something really stupid but I can’t figure what it is exactly.