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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c1">a</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c2">p</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">l</font> = <font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font>;<br/>
<a NAME="E2:27"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T84" title="INT_1:th.84">INT_1:84</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">l</font> = <font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="int_3.html#T13" title="INT_3:th.13">INT_3:13</a></i>;<br/>
<a NAME="E5:27"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">l</font>,<font color="Maroon" title="c1">a</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>;<br/>
<a NAME="E6:27"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="int_4.html#T14" title="INT_4:th.14">INT_4:14</a></i>;<br/>
<a NAME="E7:27"/>
not <font color="Maroon" title="c4">l</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="int_5.html#E5:27"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="int_5.html#E6:27"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="int_5.html#T11" target="_self" title="INT_5:th.11">Th11</a></i>;<br/>
<a NAME="E8:27"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="int_5.html#E6:27"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="int_5.html#T18" target="_self" title="INT_5:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E9:27"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="int_5.html#E4:27"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="int_5.html#T13" target="_self" title="INT_5:th.13">Th13</a></i>;<br/>


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