The third proposition shows the intuition that if the first event happens earlier than the second event, then it is not the case that the second event happens not later than the first event.

## Proposition 3.

(∀o1,o2)
earlier than(o1,o2) ⊃ ¬not later than(o2,o1)

Proof. We assume for a contradiction that there are some observations o1 and o2 such that
(∀o1,o2)
earlier than(o1,o2) ⊃ ¬not later than(o2,o1)

Then by the axiom (2.14), we have earlier than(o2,o1). Thus, earlier than(o1,o2) and earlier than(o2,o1), which by Proposition 1 implies earlier than(o1,o1). But this contradicts with Proposition 1, which says that the earlier than relation is irreflexive. ⊓⊔
Example 1. Assume the set of all possible event occurrences is {oi: 1 ≤ i ≤ 7}. The following is an example of a gso-structure, where 1. The earlier than relation is represented by a directed *acyclic* graph G1:

![0_image_0.png](0_image_0.png)

Note that in this diagram, we used the solid edges to denote the edges of the *transitive reduction*1 of the earlier than relation.