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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="E1:141"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> =  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><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="E4:141"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">B</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">B</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, <a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E5:141"/><i><font color="Green">E47</font></i>: 
(  <a href="setfam_1.html#K5">union</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> <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">B</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:141_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:141_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T57" target="_self">Th57</a>, </i>;<br/><a NAME="E5:141_1"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><b>thus </b><a NAME="E7:141_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E8:141_1"/>
<font color="Maroon">H</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, <a class="ref" href="group_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:141_1"/><b>then </b>
 ex <font color="Olive">B</font>, <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:141_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T59" target="_self">Th59</a></i>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E11:141_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E12:141_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T60" target="_self">Th60</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E15:141_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T60" target="_self">Th60</a>, </i>;<br/><b>thus </b><a NAME="E16:141_1"/>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="group_2.html#T59" target="_self">Th59</a>, , <a class="ref" href="group_2.html#T52" target="_self">Th52</a></i>;<br/><b>thus </b><a NAME="E17:141_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="group_2.html#T59" target="_self">Th59</a>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E7:141"/><b>then </b>
 ex <font color="Olive">D</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">H</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:141"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">H</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="group_2.html#T55" target="_self">Th55</a>, </i>;<br/>


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