# graph theory – Proving Easy Lemma about Flows

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.