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<b>consider </b><font color="Maroon">m'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i/>;<br/>
<a NAME="E3:17_1_1"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">m'</font> is <a href="abian.html#V1">even</a>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">n'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i/>;<br/>
<a NAME="E6:17_1_1"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">n'</font> is <a href="abian.html#V1">even</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">m''</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n''</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><font color="Maroon">m''</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n''</font></span>)</span> <a href="nat_1.html#K6">mod</a> 4 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m''</font> <a href="nat_1.html#K6">mod</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n''</font> <a href="nat_1.html#K6">mod</a> 4</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 4
<b>by </b><i><a class="ref" href="euler_2.html#T8">EULER_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n''</font> <a href="nat_1.html#K6">mod</a> 4</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 4
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 4
<b>by </b><i>, , </i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1_1_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> is <a href="pythtrip.html#V1" target="_self">square</a>
 ;<br/><b>then consider </b><font color="Maroon">mn'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:17_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mn'</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i/>;<br/><a NAME="E4:17_1_1_2"/>
<font color="Maroon">mn'</font> is <a href="abian.html#V1">even</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:17_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>


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