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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</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/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font>
 ;<br/>

<b>assume </b><a NAME="E2:42"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/>

<a NAME="E3:42"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font>
 
;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#D2" target="_self" title="HEYTING3:def.2">Def2</a></i>;<br/>
<a NAME="E5:42"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E3:42"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">m'</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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> <b> such that </b><br/><a NAME="E7:42"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c4">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">m'</font>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="heyting3.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="heyting3.html#E4:42"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self" title="HEYTING3:def.2">Def2</a></i>;<br/>
<b>thus </b><a NAME="E8:42"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E7:42"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


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