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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:16"/>
not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then consider </b><font color="Maroon">i'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="abian.html#T1">ABIAN:1</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K6">mod</a> 4 = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="int_1.html#K6">mod</a> 4
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
1 <a href="int_1.html#K6">mod</a> 4
<b>by </b><i>, <a class="ref" href="euler_1.html#T13">EULER_1:13</a></i>
<br/>.= 
1 <a href="nat_1.html#K6">mod</a> 4
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>


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