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<div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:45"/>
( not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies  not 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:45_1"/>
not 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E45</font></i>: 
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:45_1_1"/>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:45_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T80">NAT_1:80</a></i>;<br/>
<a NAME="E4:45_1_1"/><b>then </b>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<b>hence </b><a NAME="E5:45_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1"/>
not 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
( 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> implies <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>


</div>
