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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E2:41"/>
<span class="p1">(<span class="default"><span class="p2">(<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="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E3:41"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></i>;<br/>
<a NAME="E4:41"/><b>then </b>
<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="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font>,0 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<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="E5:41"/><b>then </b>
0,<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="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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">m</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/>
<a NAME="E6:41"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="int_4.html#T11" target="_self" title="INT_4:th.11">Th7</a></i>;<br/>
<a NAME="E7:41"/><b>then </b>
0,<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#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></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="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
;<br/>
<a NAME="E8:41"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></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="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<a NAME="E9:41"/>
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E31">A5</font></i>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">k</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> ) </span>}</span> ;<br/>
<a NAME="E2:41_1"/>
1 <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b>
<font color="Maroon" title="c2">m</font>,1 <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">k</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:41"/><b>then </b>
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font>

;<br/>
<a NAME="E12:41"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E8:41"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#T40" title="INT_1:th.40">INT_1:40</a></i>;<br/>
<a NAME="E13:41"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="euler_2.html#T33" title="EULER_2:th.33">EULER_2:33</a></i>;<br/>
<a NAME="E14:41"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E15:41"/><b>then </b>
<font color="Maroon" title="c2">m</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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span>,1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<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="E17:41"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font>,1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="int_4.html#T11" target="_self" title="INT_4:th.11">Th7</a></i>;<br/>
<a NAME="E18:41"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E12:41"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/>
<a NAME="E19:41"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E20:41"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></i>;<br/>


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