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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</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>assume </b><a NAME="E1:148"/><i><font color="Green">E98</font></i>: 
(  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> or  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<i><font color="Green">E99</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="group_2.html#K16">Index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T161" target="_self">Th161</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_2.html#K14">Left_Cosets</a> <span class="p2">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

;<br/>
<a NAME="E3:148"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font>, <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E4:148"/><b>then </b>
(  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="group_3.html#E1:148"><i><font color="Green">E98</font></i></a>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:148"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="group_2.html#K17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#D18">GROUP_2:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="group_3.html#E1:148"><i><font color="Green">E98</font></i></a>, <a class="txt" href="group_3.html#E3:148"><i><font color="Green">E100</font></i></a>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E8:148"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E9:148"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E2:148"><i><font color="Green">E99</font></i></a>, <a class="txt" href="group_3.html#E6:148"><i><font color="Green">E101</font></i></a></i>;<br/>


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