[proofplan]
We measure the imbalance at each vertex as outgoing flow minus incoming flow. Summing these imbalances over all vertices gives zero, because every directed edge is counted once positively at its tail and once negatively at its head. Flow conservation makes every internal vertex have zero imbalance, leaving only the source and sink imbalances. The source imbalance is $|f|$, and the negative of the sink imbalance is exactly the net inflow into $t$.
[/proofplan]
[step:Define the vertex imbalance and compute its total sum]
Define the imbalance function $b: V \to \mathbb{R}$ by assigning to each vertex $w \in V$ the real number
\begin{align*}
b(w)=
\sum_{v \in V : (w,v) \in E} f(w,v)
-
\sum_{u \in V : (u,w) \in E} f(u,w).
\end{align*}
Since $V$ and $E$ are finite, all sums below are finite. Summing over all vertices gives
\begin{align*}
\sum_{w \in V} b(w)=\sum_{w \in V}\sum_{v \in V : (w,v) \in E} f(w,v)-\sum_{w \in V}\sum_{u \in V : (u,w) \in E} f(u,w).
\end{align*}
Reindexing the two finite double sums by directed edges gives
\begin{align*}
\sum_{w \in V} b(w)=\sum_{(u,v) \in E} f(u,v)-\sum_{(u,v) \in E} f(u,v).
\end{align*}
Therefore
\begin{align*}
\sum_{w \in V} b(w)=0.
\end{align*}
The first double sum counts each edge $(u,v) \in E$ once with positive sign, namely at its tail $u$, and the second double sum counts the same edge once with negative sign, namely at its head $v$.
[guided]
The useful quantity for this first step is the net amount of flow created at a vertex. Define the imbalance function $b: V \to \mathbb{R}$ by
\begin{align*}
b(w)=
\sum_{v \in V : (w,v) \in E} f(w,v)
-
\sum_{u \in V : (u,w) \in E} f(u,w)
\end{align*}
for each $w \in V$. Thus $b(w)$ is outgoing flow from $w$ minus incoming flow into $w$. Because the directed network is finite, every sum appearing here is finite, so reindexing the sums does not require any convergence theorem.
Now sum $b(w)$ over all vertices:
\begin{align*}
\sum_{w \in V} b(w)=\sum_{w \in V}\sum_{v \in V : (w,v) \in E} f(w,v)-\sum_{w \in V}\sum_{u \in V : (u,w) \in E} f(u,w).
\end{align*}
In the first double sum, an edge $(u,v) \in E$ appears exactly when the outer vertex is its tail, namely $w=u$, so it contributes $f(u,v)$ exactly once. In the second double sum, the same edge $(u,v)$ appears exactly when the outer vertex is its head, namely $w=v$, so it contributes $f(u,v)$ exactly once with the opposite sign. Reindexing both finite sums by the edge set $E$ therefore gives
\begin{align*}
\sum_{w \in V} b(w)=\sum_{(u,v) \in E} f(u,v)-\sum_{(u,v) \in E} f(u,v)=0.
\end{align*}
This proves the total-imbalance identity. In particular, even a possible edge $(s,t)$ is counted once positively at $s$ and once negatively at $t$, so it cancels in the total sum.
[/guided]
[/step]
[step:Use conservation to isolate the source and sink imbalances]
For every internal vertex $w \in V \setminus \{s,t\}$, the assumed flow-conservation identity gives
\begin{align*}
\sum_{v \in V : (w,v) \in E} f(w,v)
=
\sum_{u \in V : (u,w) \in E} f(u,w),
\end{align*}
so $b(w)=0$. Hence
\begin{align*}
0=\sum_{w \in V} b(w).
\end{align*}
Splitting the finite sum into the source, the sink, and the internal vertices gives
\begin{align*}
\sum_{w \in V} b(w)=b(s)+b(t)+\sum_{w \in V \setminus \{s,t\}} b(w).
\end{align*}
Since every term in the last sum is zero, we obtain
\begin{align*}
0=b(s)+b(t).
\end{align*}
Therefore $b(s)=-b(t)$.
[/step]
[step:Identify the two remaining imbalances with the claimed quantities]
By the definition of the flow value,
\begin{align*}
b(s)
=
\sum_{v \in V : (s,v) \in E} f(s,v)
-
\sum_{u \in V : (u,s) \in E} f(u,s)
=
|f|.
\end{align*}
Also,
\begin{align*}
-b(t)=-\left(\sum_{v \in V : (t,v) \in E} f(t,v)-\sum_{u \in V : (u,t) \in E} f(u,t)\right).
\end{align*}
Distributing the minus sign gives
\begin{align*}
-b(t)=\sum_{u \in V : (u,t) \in E} f(u,t)-\sum_{v \in V : (t,v) \in E} f(t,v).
\end{align*}
Since $b(s)=-b(t)$, substituting these two identities gives
\begin{align*}
|f|
=
\sum_{u \in V : (u,t) \in E} f(u,t)
-
\sum_{v \in V : (t,v) \in E} f(t,v).
\end{align*}
This is the desired source-sink flow balance.
[/step]
