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

<b>let </b><font color="Maroon">c<sub>1</sub></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">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:24"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>assume </b><a NAME="E4:24"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E5:24"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E6:24"/><i><font color="Green">E20</font></i>: 
(  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font>, <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E7:24"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E1:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="group_8.html#E3:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="group_8.html#E4:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="group_8.html#E5:24"><i><font color="Green">E19</font></i></a>, <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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></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">E22</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></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">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E7:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_2.html#T179">GROUP_2:179</a></i>;<br/>
<a NAME="E10:24"/><i><font color="Green">E23</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:24"><i><font color="Green">E22</font></i></a>, <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">c<sub>3</sub></font>, <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="group_8.html#E6:24"><i><font color="Green">E20</font></i></a>, <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">E24</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E10:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E25</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E12:24"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E14:24"/><i><font color="Green">E26</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></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">c<sub>8</sub></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">E27</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E13:24"><i><font color="Green">E25</font></i></a>, <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">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></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">E15</font></i></a>, <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">c<sub>8</sub></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">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E3:24_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:24_1_1"><i><font color="Green">E28</font></i></a>, <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">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:24_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E2:24_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="group_8.html#E2:24_1_1_1"><i><font color="Green">E29</font></i></a>, <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">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">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">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>thus </b><a NAME="E6:24_1_1"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E2:24_1_1"><i><font color="Green">E27</font></i></a></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">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</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>
<a NAME="E15:24"/><i><font color="Green">E27</font></i>: 
 <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:24"><i><font color="Green">E16</font></i></a>, <a class="txt" href="group_8.html#E3:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="group_8.html#E7:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_8.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E28</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></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">E29</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:24_2"/>
<font color="Maroon">c<sub>3</sub></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">E15</font></i></a>, <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">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E4:24_2"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E14:24"><i><font color="Green">E26</font></i></a>, <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="txt" href="group_8.html#E15:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="group_8.html#E1:24_2"><i><font color="Green">E29</font></i></a>, <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">E29</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:24"><i><font color="Green">E22</font></i></a>, <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">c<sub>4</sub></font>, <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="group_8.html#E6:24"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E19:24"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font>, <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></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">E30</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="int_1.html#R2">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E17:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green">E31</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></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">c<sub>8</sub></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">E32</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E20:24"><i><font color="Green">E30</font></i></a>, <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">c<sub>8</sub></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">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:24_3_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E3:24_3_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></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">E15</font></i></a>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E4:24_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:24_3_1_1"><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">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E6:24_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E2:24_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="group_8.html#E2:24_3_1_1"><i><font color="Green">E33</font></i></a>, <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">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">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">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>thus </b><a NAME="E5:24_3_1"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E2:24_3_1"><i><font color="Green">E32</font></i></a></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">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="group_2.html#K17">index</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/>


</div><b>end;</b></div>
<a NAME="E22:24"/><i><font color="Green">E32</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></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<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:24"><i><font color="Green">E16</font></i></a>, <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">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E4:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="group_8.html#E7:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_8.html#T19" target="_self">Th19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:24"/>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></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">E33</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:24_5"/>
<font color="Maroon">c<sub>4</sub></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">E15</font></i></a>, <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">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E4:24_5"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E21:24"><i><font color="Green">E31</font></i></a>, <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="txt" href="group_8.html#E22:24"><i><font color="Green">E32</font></i></a>, <a class="txt" href="group_8.html#E1:24_5"><i><font color="Green">E33</font></i></a>, <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<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E16:24"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
