<?xml version="1.0"?>
<div class="add">

<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="Red">::  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><br/> for <font color="Olive" title="b2">b</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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; <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">p</font></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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <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; <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font> holds <br/> <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</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="Red">::  thesis: </font></span></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; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font> implies  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E28">A2</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="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></span><br/>

<a NAME="E4:36"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font>,<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> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="int_5.html#T28" target="_self" title="INT_5:th.28">Th28</a></span>;<br/>
<a NAME="E5:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E6:36"/>
<font color="Maroon" title="c3">b</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="int_4.html#T14" title="INT_4:th.14">INT_4:14</a></span>;<br/>
<a NAME="E7:36"/><span class="kw">then </span>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</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> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="int_5.html#T28" target="_self" title="INT_5:th.28">Th28</a></span>;<br/>
<a NAME="E8:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></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="c3">b</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E9:36"/>
<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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E10:36"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font></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"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="int_5.html#T13" target="_self" title="INT_5:th.13">Th13</a></span>;<br/>
<a NAME="E11:36"/><span class="kw">then </span>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font>, <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E12:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<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"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:36_1"/>
(  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></span><br/><div class="add"><a NAME="E2:36_1_1"/><span class="kw">then </span>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_2"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_1_2_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_2_1"/><span class="kw">then </span>
<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> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E1:36_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a></span>;<br/><a NAME="E3:36_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:36_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:36_1_2"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
