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<span class="kw">let </span><font color="Maroon" title="c1">a</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_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>,<font color="Olive" title="b1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>



<span class="kw">assume </span><a NAME="E1:31"/>
<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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:31"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" title="INT_1:th.62">INT_1:62</a></span>;<br/>
<a NAME="E3:31"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T7" target="_self" title="INT_5:th.7">Th7</a></span>;<br/>
<a NAME="E4:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" title="INT_1:th.62">INT_1:62</a></span>;<br/>
<a NAME="E5:31"/>
<font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T9" target="_self" title="INT_5:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#D3" target="_self" title="INT_5:def.3">Def3</a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E26">A1</font></span></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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