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<div class="add">

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

<a NAME="E1:35"/><i><font color="Green">E25</font></i>: 
( 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<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="E3:35_1"/><i><font color="Green">E26</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="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:35_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="pepin.html#E3:35_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> implies 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/>

<a NAME="E2:35_2"/><b>then </b>
<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_2.html#T23">NAT_2:23</a></i>;<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="E4:35_2"/><i><font color="Green">E26</font></i>: 
( <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 &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>thus </b><a NAME="E5:35_2"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E4:35_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
( 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="txt" href="pepin.html#E1:35"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
