Order Theory/Total orders

Definition (total order):

An order on a set is said to be total if and only if for each , exactly one of the possibilities , or occurs.

{{proposition|series order induced by total orders is total|Whenever is a totally ordered set

Proposition (lexicographic order induced by total orders is total):

Whenever is a well-ordered set and are totally ordered sets, the lexicographic order on is total.

Proof: Let any two elements and of be given. Then either , or there exists a smallest so that . Since is total, either or , and thus either or .