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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:74"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:74_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_2_1_1"/>
(  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_2_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_2_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:74_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_1.html#E1:74_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="group_1.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_2_1_1_2"/><i><font color="Green">E62</font></i>: 
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E3:74_1_1_2_1_1_2"/><b>then </b>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:74_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_1.html#E1:74_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="group_1.html#D12" target="_self">Def12</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:74_1_1_2_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><i><font color="Green">E64</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E5:74_1_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="group_1.html#T48" target="_self">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82" target="_self">Th82</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5" target="_self">Def5</a></i>
;<br/><a NAME="E7:74_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:74_1_1_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_1.html#E5:74_1_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E8:74_1_1_2_1_1_2"/><b>then </b>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:74_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_1.html#E2:74_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_1.html#E6:74_1_1_2_1_1_2"><i><font color="Green">E64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:74_1_1_2_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:74_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_1.html#E5:74_1_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="group_1.html#E10:74_1_1_2_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>hence </b><a NAME="E12:74_1_1_2_1_1_2"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1_1_2"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:74"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="group_1.html#E1:74"><i><font color="Green">E58</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:74"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div>
