I have one question please about proving that the flow $f(χ) $ across any cut is equal to the flow value of the network $|f|$. Now, the flow of the cut is the difference between flow of forward edges (start if $V_s$ and ends in $V_t$), where $s$ and $t$ are source and sink vertices and $χ$ is the cut of the vertices into $V_s$ and $V_t$.
If we take total flow of a cut, which is defined as the flow $F$ as follows,
$$F = \sum_{v\in V_s}\left(\sum_{e\in E^{+}(v)}^{}(f(e))- \sum_{e\in E^{-}(v)}^{}(f(e))\right)$$
Case 1: for all vertices other than the source, we will have by conservation law that, for all vertices $v \in V_s$ other than source $s$,
$$F = \sum_{v\in V_s}\left(\sum_{e\in E^{+}(v)}^{}(f(e))- \sum_{e\in E^{-}(v)}^{}(f(e))= 0\right) = |f| $$
where $|f|$ is the flow of a network as the total flow from the source.
Case 2: On the other hand, for each edge e that is not a forward or a backward edge of cut χ, the sum F contains both the term $f(e)$ and the term $−f(e)$, which cancel each other, or neither the term $f(e)$ nor the term $−f(e)$. Thus, $F = f(χ)$, where $f(χ)$ is the flow a cross the cut that is the difference between flow of forward edges minus flow of backward edges.
Problem: can you please clarify case 2 as I don't get it clearly?