Editing Mathematical proof (section)

===Direct proof===
{{Main|Direct proof}}

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.<ref>Cupillari, p. 20.</ref> For example, direct proof can be used to prove that the sum of two [[parity (mathematics)|even]] [[integer]]s is always even:

:Consider two even integers ''x'' and ''y''. Since they are even, they can be written as ''x''&nbsp;=&nbsp;2''a'' and ''y''&nbsp;=&nbsp;2''b'', respectively, for some integers ''a'' and ''b''. Then the sum is ''x''&nbsp;+&nbsp;''y''&nbsp;= 2''a''&nbsp;+&nbsp;2''b''&nbsp;= 2(''a''+''b''). Therefore ''x''+''y'' has 2 as a [[divisor|factor]] and, by definition, is even. Hence, the sum of any two even integers is even.

This proof uses the definition of even integers, the integer properties of [[Closure (mathematics)|closure]] under addition and multiplication, and the [[distributive property]].