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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">C</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">C</font>, <font color="Maroon">B</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><a NAME="E2:24"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">D</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">D</font> <a href="group_2.html#R1">=</a> <font color="Maroon">A</font> <a href="group_2.html#K9">/\</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">E</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E4:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">E</font> <a href="group_2.html#R1">=</a> <font color="Maroon">A</font> <a href="group_2.html#K9">/\</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E5:24"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="group_2.html#R1">=</a> <font color="Maroon">A</font> <a href="group_2.html#K9">/\</a> <font color="Maroon">B</font>
 ;<br/>

<b>assume </b><a NAME="E6:24"/><i><font color="Green">E48</font></i>: 
(  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K17">index</a> <font color="Maroon">A</font>, <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E7:24"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">C</font> is <a href="group_1.html#V6">finite</a> &amp; <font color="Maroon">F</font> <a href="group_2.html#R1">=</a> <font color="Maroon">E</font> &amp; <font color="Maroon">F</font> <a href="group_2.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, , , , <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T179">GROUP_2:179</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T179">GROUP_2:179</a></i>;<br/>
<a NAME="E10:24"/><i><font color="Green">E64</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E11:24"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font>, <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i>, <a class="ref" href="euler_2.html#T1">EULER_2:1</a></i>;<br/>
<a NAME="E12:24"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E68</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:24"/><i><font color="Green">E69</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E3:24_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">C</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E4:24_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:24_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 
;<br/>
<a NAME="E3:24_1_1_1"/>
<font color="Maroon">A</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E4:24_1_1_1"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:24_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b><font color="Maroon">n</font> = <font color="Maroon">i</font>;<br/>

<b>thus </b><a NAME="E6:24_1_1"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</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>
<a NAME="E15:24"/><i><font color="Green">E74</font></i>: 
 <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E75</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E76</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:24_2"/>
<font color="Maroon">A</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:24_2"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:24_2"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<b>hence </b><a NAME="E5:24_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:24"/><i><font color="Green">E77</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">B</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E18:24"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font>, <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E19:24"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font>, <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="euler_2.html#T1">EULER_2:1</a></i>;<br/>
<a NAME="E20:24"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green">E81</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<a NAME="E1:24_3"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E2:24_3_1"/><i><font color="Green">E82</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E3:24_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_3_1_1"/>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:24_3_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 
;<br/>
<a NAME="E3:24_3_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">C</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E4:24_3_1_1"/>
<font color="Maroon">B</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:24"><i><font color="Green">E34</font></i></a>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E5:24_3_1_1"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">E</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:24_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b><font color="Maroon">n</font> = <font color="Maroon">i</font>;<br/>

<b>thus </b><a NAME="E5:24_3_1"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_3"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</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>
<a NAME="E22:24"/><i><font color="Green">E85</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

<a NAME="E1:24_4"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="group_4.html#T74">GROUP_4:74</a></i>;<br/>
<b>hence </b><a NAME="E2:24_4"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:24"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_5"/><i><font color="Green">E87</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 ;<br/>

<a NAME="E2:24_5"/>
<font color="Maroon">B</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:24_5"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">E</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:24_5"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<b>hence </b><a NAME="E5:24_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:24"/>
(  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">A</font>,<font color="Maroon">D</font> &amp;  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">B</font>,<font color="Maroon">E</font> )
 <b>by </b><i/>;<br/>


</div>
