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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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E30</font></i>: 
 <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E2:42"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:42"/><b>then </b><i><font color="Green">E31</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E4:42"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:42"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">m'</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:42"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m'</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E8:42"/>
<font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E1:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:42"/>
contradiction
 ;<br/>


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