**Proof**: *(related to "Comparing the Elements of Strictly Ordered Sets")*

- Let $(V,\prec)$ be a strictly ordered set.
- By definition, $”\prec”$ is a total order. Therefore it is ensured that
*at least one*of the possibilities $a \prec b,$ $a \succ b,$ or $a=b$ is fullfilled. It remains to be shown that*exactly one*is possible:- The possibilities $a \prec b$ and $a=b$ cannot occur simultaneously, because $”\prec”$ is irreflexive, by definition.
- The same argument holds for the possibilities $a \succ b$ and $a=b$ occuring simultaneously.
- Neither can the possibilities $a \succ b$ and $a \prec b$ occur simultaneously, because $”\prec”$ is asymmetric, by definition.

q.e.d

| | | | created: 2019-02-03 09:34:07 | modified: 2019-02-03 09:34:48 | by: *bookofproofs* | references: [979], [8055]

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[8055] **Hoffmann, D.**: “Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise”, Hoffmann, D., 2018

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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