 The reader should consider what happens when \(R\) is taken to be the relation (immediately) precedes. Appealing to our intuitive grasp of the numbers, we can say that it is an instance of Fact (1) that if 10 precedes 11, then 10 precedes* 11. Moreover, precedes is a witness to Fact 2: that 10 precedes* 12 does not imply that 10 precedes 12. The transitivity of precedes* is an instance of Fact (7). Below, when we restrict ourselves to the natural numbers, it becomes intuitive to think of the difference between precedes and precedes* as the difference between immediately precedes and less-than. 