exists a critical item $c$ in $KP^{LP}(x^*)$, then $z(CRIT_1(c))$ is a valid lower bound on $z^*$.

Under the assumption that item $c$ is critical in $KP^{LP}(x^*)$, the optimal BKP solution $x^*$ constitutes a feasible solution for model $CRIT_1(c)$. Let denote by $z_1$ the corresponding solution value that coincides with the value of the split solution in $KP(x^*)$. Since the follower maximizes the profits in $KP(x^*)$ obtaining a solution with a value greater than (or equal to) the one of the split solution, we have $z_1 \leq z^*$. But this means that there exists an optimal solution of model $CRIT_1(c)$ such that $z(CRIT_1(c)) \leq z_1$ which implies a lower bound on $z^*$.

The previous proposition already provides a first significant lower bound for the problem. However, following the reasoning in the proof of Proposition \[firstProp\], we remark that improved bounds on $z^*$ can be derived by considering any feasible solution for $KP(x^*)$ that might be obtained by removing (adding) items that were not interdicted by the leader and that were selected (not selected) by the split solution, provided that the follower capacity is not exceeded. Indeed, this corresponds to removing tuples of items $i \in [1,c-1]: x_i=0$ and/or to adding tuples 