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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">p</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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> </span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E23">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="ref" href="int_5.html#T19" target="_self" title="INT_5:th.19">Th19</a></i>;<br/>
<a NAME="E3:27"/>
( <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<a NAME="E4:27"/><b>then </b>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <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="txt" href="#E2:27"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="int_3.html#T10" title="INT_3:th.10">INT_3:10</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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>
<font color="Maroon" title="c2">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="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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/>
<a NAME="E8:27"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">p</font> <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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E9:27"/><b>then </b>
<font color="Maroon" title="c2">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="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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="int_5.html#T2" target="_self" title="INT_5:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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