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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">I</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<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>let </b><font color="Maroon">J</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">H</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:143"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:143"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E3:143"/><i><font color="Green">E47</font></i>: 
(  <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> &amp;  <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">I</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">I</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, </i>;<br/>
<a NAME="E4:143"/>
<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="E5:143"/><b>then </b>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">J</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">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:143"/><b>then </b>
( <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">J</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">I</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">I</font></span>)</span> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, , <a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<b>hence </b><a NAME="E7:143"/>
 <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">J</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="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>


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