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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" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:149"/><i><font color="Green">E97</font></i>: 
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:149"/><i><font color="Green">E98</font></i>: 
 <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K8">Ord</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:149"/>
(  <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> &amp;  <a href="group_1.html#K8">Ord</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> &amp;  <a href="group_1.html#K8">Ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="group_2.html#K16" target="_self">Index</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E2:149"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E4:149"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="group_2.html#E1:149"><i><font color="Green">E97</font></i></a>, <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">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></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<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="E7:149"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <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_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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="E9:149"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E1:149"><i><font color="Green">E97</font></i></a>, <a class="ref" href="group_2.html#D18" target="_self">Def18</a></i>;<br/>
<b>thus </b><a NAME="E10:149"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#R1" target="_self">=</a>  <a href="group_2.html#K5" target="_self">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E2:149"><i><font color="Green">E98</font></i></a>, <a class="txt" href="group_2.html#E5:149"><i><font color="Green">E99</font></i></a>, <a class="txt" href="group_2.html#E7:149"><i><font color="Green">E100</font></i></a>, <a class="txt" href="group_2.html#E9:149"><i><font color="Green">E101</font></i></a>, <a class="ref" href="group_2.html#T184" target="_self">Th184</a></i>;<br/>


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