Flow value is at most cut capacity

Statement

Let $G$ be a flow network on a finite directed graph. It has a source $s$, a sink $t$, and a capacity $c(e) \ge 0$ on each edge $e$.

Let $\phi$ be a flow in $G$. That is, $0 \le \phi(e) \le c(e)$ for each edge $e$, and at each vertex other than $s$ and $t$ the flow in equals the flow out.

Let $V = L \cup U$ be a cut. That is, $L$ and $U$ are disjoint, $s \in L$ and $t \in U$. The capacity of the cut is the sum of $c(e)$ over the edges that go from $L$ to $U$.

Then the value of $\phi$ is at most the capacity of the cut.