How can we derive this lower bound of a special cut in a graph?
Consider a cut $C$ in $G^\prime$. Let $c_k(C)$ be its uniform cut capacity in $G$. Then $c_k(C) \geq c_k(s,t)$. By definition of $c_k(C)$, exactly $k$ packets, each of size $c_k(C)/k$ can be packed into the cut $C$. So sum of $\lfloor ku_e /c_k(C) \rfloor$ over the cut is at least $k$. As $c_k(C) \geq c_k(s,t)$, $ku_e /c_k(C) \leq ku_e /c_k(s,t)$. So 

$$
\sum_C u_e^\prime = \sum_C \lfloor ku_e /c_k(s,t)\rfloor \geq \sum_C \lfloor ku_e /c_k(C)\rfloor \geq k
$$