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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2
;<br/>

<b>set </b><font color="Maroon">k</font> = <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2;<br/>
<a NAME="E2:1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E2:1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_1_2"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/>


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