x_i). 
$$

Correspondingly, a new model can be generated by introducing a non-negative variable $\pi$ that carries the maximum additional profit to the split solution value provided by any of the additional constraints (\[TupleCons\]) indicated above. These constraints, denoted as $\mathcal{F}(\pi, x, k)$, link variable $\pi$ to variables $x_i$ and $k_j$. The model (denoted as $CRIT_2(c)$) is as follows.\
[$\quad CRIT_2(c)$:]{} $$\begin{aligned}
\text{min}\quad & \sum\limits_{i=1}^{c-1} p_{i}(1-x_i) + \pi \label{eq:ObjCRIT2}\\
    \text{subject to}\quad
    & \mathcal{F}(\pi,x,k) \label{genConstonPi2} \\
    & \eqref{eq:capLeadCRIT}, \eqref{eq:varDefKCRIT} \tag*{} \\ 
    & \pi \geq 0 \label{eq:varDefPi}  \end{aligned}$$

Clearly, due to the addition of constraints in $\mathcal{F}(\pi,x,k)$, for any $c$ we have $z(CRIT_1(c)) \leq  z(CRIT_2(c))$. Notice that, in all these additional constraints, only items which will not be interdicted by the leader can be packed and the follower’s capacity constraint is not violated. We denote as *proper* any set $\mathcal{F}(\pi,x,k)$ that satisfies both conditions. After the set $\mathcal{F}(\pi, x, k)$ is built, variable $\pi$ will carry the maximum profit obtainable in addition to the profit of the split solution.

\[secTh\] If $KP^{LP}(x^*)$ admits a critical item $c$ and model $CRIT_2(c)$ has a proper set $\mathcal{F}(\pi, x, k)$, then $z(CRIT_2(c)) \leq z^*$.

Since model $CRIT_2(c)$ considers feasible solutions for $KP(x^*)$, the inequality holds 