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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:20"/>
not <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ;<br/>

<a NAME="E4:20"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E15"><i><font color="Green" title="E13">Lm2</font></i></a>, <a class="ref" href="int_2.html#T12" title="INT_2:th.12">INT_2:12</a></i>;<br/>
<a NAME="E5:20"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:20"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="int_5.html#T1" target="_self" title="INT_5:th.1">Th1</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">a</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="int_5.html#T12" target="_self" title="INT_5:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:20"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>


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