Editing Mathematical proof (section)

=== Closed chain inference ===
{{Main|Closed chain inference}}

A closed chain inference shows that a collection of statements are pairwise equivalent.

In order to prove that the statements <math>\varphi_1,\ldots,\varphi_n</math> are each pairwise equivalent, proofs are given for the implications <math>\varphi_1\Rightarrow\varphi_2</math>, <math>\varphi_2\Rightarrow\varphi_3</math>, <math>\dots</math>, <math>\varphi_{n-1}\Rightarrow\varphi_n</math> and <math>\varphi_{n}\Rightarrow\varphi_1</math>.<ref>{{Cite book |last1=Plaue |first1=Matthias |url=https://books.google.com/books?id=-WCHDwAAQBAJ |title=Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis |last2=Scherfner |first2=Mike |date=2019-02-11 |publisher=Springer-Verlag |isbn=978-3-662-58352-4 |pages=26 |language=de |trans-title=Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis}}</ref><ref>{{Cite book |last1=Struckmann |first1=Werner |url=https://books.google.com/books?id=1epNDQAAQBAJ |title=Mathematik für Informatiker: Grundlagen und Anwendungen |last2=Wätjen |first2=Dietmar |date=2016-10-20 |publisher=Springer-Verlag |isbn=978-3-662-49870-5 |pages=28 |language=de |trans-title=Mathematics for Computer Scientists: Fundamentals and Applications}}</ref>

The pairwise equivalence of the statements then results from the [[Transitive relation|transitivity]] of the [[material conditional]].