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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  st <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b1">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="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies <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>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<span class="kw">assume </span><a NAME="E3:18"/>
 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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_5.html#E15"><span class="lab"><font color="Green" title="E13">Lm2</font></span></a>, <a class="ref" href="int_2.html#T2" title="INT_2:th.2">INT_2:2</a></span>;<br/>
<a NAME="E5:18"/>
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="int_5.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<a NAME="E6:18"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="int_5.html#T1" target="_self" title="INT_5:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T12" target="_self" title="INT_5:th.12">Th12</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:18"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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