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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:34"/><i><font color="Green" title="E23">A1</font></i>: 
( 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:34_1"/>
<font color="Maroon" title="c1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="pepin.html#E3:34_1"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34"/>
( <font color="Maroon" title="c1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon" title="c1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<a NAME="E2:34_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_2.html#T23" title="NAT_2:th.23">NAT_2:23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:34_2"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:34_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E4:34_2"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
( 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> iff <font color="Maroon" title="c1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="pepin.html#E1:34"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>


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