 Proof: If \(i_1, i_2 \in e'\) and \(i_1 \lt i \lt i_2\), it has to be shown that \(e \in i\), too. Suppose it is not.  Then there is \(d \in i\) such that \(\neg O(d, e)\), hence \(P(d, e)\) or \(P(e, d)\). In the first case \(i \lt i_1\) and in the second case \(i_2 \lt i\). Both yield a contradiction.\(\Box\) 