The strictness of IRS can be obtained by just removing the label from , and therefore by increasing by one at most its compactness (or linearity compactness). Because a cyclic interval is the union of two linear intervals, we have the trivial collection of inclusions:
In the following, means that and . Because in [FJ88] and in [BvLT91] it is shown - but not in k- , we have:
Theorem 57, and 58 can be improved by Theorem 47.
From Theorem 47, for every , - - , and also - . From the second point of Theorem 47, - implies that . Hence - . It follows that k- - and k- - . Similarly - , but - , and - .