I am learning algorithms to solve Maximum Flow problem by reading the CLRS book and confused by the following figure:
That is:
A flow in a residual network provides a roadmap for adding flow to the original flow network. If $f$ is a flow in $G$ and $f'$ is a flow in the corresponding residual network $G_f$, we define $f \uparrow f'$, the augmentation of flow $f$ by $f'$, to be a function from $V \times V$ to $R$, defined by
$$(f \uparrow f')(u, v) = \begin{cases} f(u,v) + f'(u, v) - f'(v, u) & > \text{if (u,v) $\in$ E} \\ 0 & \text{otherwise} \end{cases}$$
How the flow network in (c), for example $(s, v_2)$ got the flow 12 ? If we follow the formula, it must have a flow 5: $8 + 5 - 8 = 5$