<?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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  holds <br/>(  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">a</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Olive" title="b1">n</font> or <font color="Maroon" title="c1">a</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Olive" title="b1">n</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</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"> (  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">a</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c1">a</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">cLa</font> = <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">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:46"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E3:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A1</font></span>: 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T5" title="PEPIN:th.5">PEPIN:5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:46"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c1">a</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<a NAME="E5:46"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">cLa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" title="NAT_D:th.67">NAT_D:67</a></span>;<br/>
<a NAME="E6:46"/><span class="kw">then </span>
<font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">cLa</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="nat_d.html#T64" title="NAT_D:th.64">NAT_D:64</a></span>;<br/>
<a NAME="E7:46"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">cLa</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E8:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E9:46"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
;<br/>
<a NAME="E10:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</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#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<a NAME="E11:46"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<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">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<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">n</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"> contradiction</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><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">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" title="NAT_D:th.67">NAT_D:67</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="int_1.html#T62" title="INT_1:th.62">INT_1:62</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:46_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E26">A2</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/>


</div><span class="kw">end;</span></div>
<a NAME="E12:46"/>
<font color="Maroon" title="c3">cLa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="nat_d.html#T62" title="NAT_D:th.62">NAT_D:62</a></span>;<br/>
<a NAME="E13:46"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">cLa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/>
<a NAME="E14:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
<a NAME="E15:46"/>
<font color="Maroon" title="c3">cLa</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T65" title="NAT_D:th.65">NAT_D:65</a></span>;<br/>
<a NAME="E16:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">cLa</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T64" title="NAT_D:th.64">NAT_D:64</a></span>;<br/>
<a NAME="E17:46"/>
<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">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="nat_d.html#T62" title="NAT_D:th.62">NAT_D:62</a></span>;<br/>
<a NAME="E18:46"/><span class="kw">then </span>
<font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E19:46"/><span class="kw">then </span>
( <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> or <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">cLa</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></span>;<br/>
<a NAME="E20:46"/><span class="kw">then </span>
( <font color="Maroon" title="c3">cLa</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c3">cLa</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:46"/>
( <font color="Maroon" title="c1">a</font>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c1">a</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="int_1.html#T15" title="INT_1:th.15">INT_1:15</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/>


</div>
