<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:22"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/>
<a NAME="E4:22_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="txt" href="nat_2.html#E3:22_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:22"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:22"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/>


</div>
