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<b>consider </b><font color="Maroon" title="c3">m'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/>
<a NAME="E3:17_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
not <font color="Maroon" title="c3">m'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E2:17_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="pythtrip.html#T2" target="_self" title="PYTHTRIP:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">n'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/>
<a NAME="E6:17_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
not <font color="Maroon" title="c4">n'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E5:17_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="pythtrip.html#T2" target="_self" title="PYTHTRIP:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">m''</font> = <font color="Maroon" title="c1">m</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">n''</font> = <font color="Maroon" title="c2">n</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green" title="E12">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n''</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m''</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n''</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4
<b>by </b><i><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n''</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4
<b>by </b><i><a class="txt" href="pythtrip.html#E2:17_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="pythtrip.html#E3:17_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="pythtrip.html#T4" target="_self" title="PYTHTRIP:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4
<b>by </b><i><a class="txt" href="pythtrip.html#E5:17_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="pythtrip.html#E6:17_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="pythtrip.html#T4" target="_self" title="PYTHTRIP:th.4">Th4</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1_1_2"/>
<font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">mn'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:17_1_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">mn'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/><a NAME="E4:17_1_1_2"/>
<font color="Maroon" title="c7">mn'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:17_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="pythtrip.html#T2" target="_self" title="PYTHTRIP:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E5:17_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pythtrip.html#E9:17_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="pythtrip.html#E3:17_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="pythtrip.html#T3" target="_self" title="PYTHTRIP:th.3">Th3</a></i>;<br/></div>
<b>end;</b></div>


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