Totally ordered partition of a set

[jason17349]

If I have a totally ordered set and then create a noncrossing partition of that set it seems intuitively obvious that each block of the partition would be totally ordered as well. Can I assume this inheritance or do I need to prove each block is totally ordered? How would one go about proving that if it is the case.

 

[micromass]

In general, if X is totally ordered and if $A\subseteq X$, then A is totally ordered.
The proof is not difficult, just use the definition of total order.