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Order Theory/Total orders
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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
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.
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 .