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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_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:125"/>
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<a NAME="E2:125"/><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="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E3:125"/><b>then </b>
(  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/>
<b>hence </b><a NAME="E4:125"/>
(  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a> &amp;  <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>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


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