Let \(\theta = (N, L, p, r, d, \mathscr {D}) \in \Theta \) be an MCL situation for which there exist for each \(i \in N\) a \(k\in L\) and \(m \in L\) with \(k\le m\) such that \(d_{ij} \le \mathscr {D}\) for all \(j \in L\) with \(k \le j \le m\) and \(d_{ij} > \mathscr {D}\) otherwise. Let \(N^* = N \cup \{n+1\}\). Now, consider matrix \(B = (b_{ij})_{i \in N^*, j \in L}\) with \( b_{ij}= 1\) for all \(i \in N\) and all \(j \in L\) with \(d_{ij} \le \mathscr {D}\) and \(b_{ij} = 0\) for all \(i \in N\) and all \(j \in L\) for which \(d_{ij}> \mathscr {D}\) and with \(b_{n+1,j} = 1\) for all \(j \in L\). So, for each \(i \in N^*\), there exist a \(k\in L\) and \(m \in L\) with \(k\le m\) such that \(b_{ij} = 1\) for all \(j \in L\) for which \(k \le j \le m\) and \(b_{ij} =0\) otherwise. Hence, matrix *B* has the consecutive ones property, and based on Theorem 11.8, is also totally unimodular. Let \(\overline{B}=(\overline{b}_{ij})_{i \in N^*,j \in L}\) be given by \(\overline{b}_{ij}=-{b}_{ij}\) for all \(i \in N\) and all \(j \in L\) and \(\overline{b}_{n+1,j} = b_{n+1,j}\) for all \(j \in L\). Based on Theorem 11.7 (*i*), matrix \(\overline{B}\) is totally unimodular (as well) as it results from matrix *B* by multiplying the first *n* rows by \(-1\). Now, let \(\widehat{B}=(\widehat{b}_{ij})_{i \in N^*, j \in N \cup L}\) be given by \(\widehat{b}_{ij}=\overline{b}_{ij}\) for all \(i \in N^*\) and all \(j \in L\), \(\widehat{b}_{ij} = 1\) for all \(i \in N^*\) and all \(j \in N\) for which \(i=j\), and \(\widehat{b}_{ij}=0\) for all \(i \in N^*\) and \(j \in N\) for which \(i \not =j\). Based on Theorem 11.7 (*ii*), matrix \(\widehat{B}\) is also totally unimodular as it results from matrix \(\overline{B}\) by adding *n* columns, each of them consisting of zeros only, except for one entry with value 1, which varies from the first position (in the first column) to the *n*-th position (in the *n*-th column). Now, let \(N^{**} = \{n+2,n+3,\ldots , 2n+1\}\) and matrix \(\widetilde{B} = ((\widetilde{b}_{ij})_{i \in N^* \cup N^{**}, j \in N \cup L}\) be given by \(\widetilde{b}_{ij} = \widehat{b}_{ij}\) for all \(i \in N^*\) and all \( j \in N \cup L\) and \(\widetilde{b}_{ij} = 1\) for all \(i \in N^{**}\) and all \(j \in N \cup L\) for which \(i=j+n+1\) and \(\widetilde{b}_{ij} =0\) for all \(i \in N^{**}\) and all \(j \in N \cup L\) for which \(i \not =j+n+1\). Based on Theorem 11.7 (*ii*), matrix \(\widetilde{B}\) is totally unimodular as it results from matrix \(\widehat{B}\) by adding *n* rows, each of them consisting of zeros only, except for one entry with value 1, which varies from the first position (in the first row) to the *n*-th position (in the *n*-th row). Now, observe that \(\widetilde{B} = A\), where *A* represents the matrix of the linear programming formulation \(RMCL^{\theta }(N)\) in standard form (i.e., in the form \(Ax \le b\)). As the optimum of the linear programming formulation \(RMCL^{\theta }(N)\) is finite, vector *b* has all integer entries, and \(p_i \in \mathbb {R}_+\) for all \(i \in N\), it follows, based on Theorem 11.6, that \(RMCL^{\theta }(N)\) has integer optimal solutions. As \(0 \le y_i \le 1\) for all \(i \in N\), it follows that there always exist optimal solutions of \(RMCL^{\theta }(N)\) for which \(0 \le x_j \le 1\) for all \(j \in L\). So, \(RMCL^{\theta }(N)\) has an integer optimal solution with not only \(y_i \in \{0,1\}\) for all \(i \in N\), but also \(x_j \in \{0,1\}\) for all \(j \in L\). This implies that \(\mathbf{opt}(RMCL^{\theta }(N)) = \mathbf{opt}(MCL^{\theta }(N))\) and so, based on Theorem 11.4, the core of \((N,v^{\theta })\) is non-empty, which concludes this proof. \(\square \)