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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:149"/><i><font color="Green">E35</font></i>: 
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:149"/><i><font color="Green">E36</font></i>: 
 <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K8">Ord</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:149"/>
(  <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="group_1.html#K8">Ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_1.html#K8">Ord</a> <font color="Maroon">G</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E4:149"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/>
<b>hence </b><a NAME="E5:149"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>

<b>then consider </b><font color="Maroon">C</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:149"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:149"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E10:149"/>
<font color="Maroon">H</font> <a href="group_2.html#R1" target="_self">=</a>  <a href="group_2.html#K5" target="_self">(1).</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_2.html#T57" target="_self">Th57</a>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T60" target="_self">Th60</a>, <a class="ref" href="group_2.html#T53" target="_self">Th53</a></i>;<br/>


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