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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">J</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:143"/><i><font color="Green" title="E84">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 <b>and </b><br/><a NAME="E2:143"/><i><font color="Green" title="E85">A2</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">J</font>
 ;<br/>

<a NAME="E3:143"/><i><font color="Green" title="E86">A3</font></i>: 
(  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">H</font></span>)</span> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">I</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_2.html#E1:143"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="group_2.html#T177" target="_self" title="GROUP_2:th.177">Th177</a></i>;<br/>
<a NAME="E4:143"/>
<font color="Maroon" title="c3">H</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_2.html#E1:143"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="group_2.html#T48" target="_self" title="GROUP_2:th.48">Th48</a></i>;<br/>
<a NAME="E5:143"/><b>then </b>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T177" target="_self" title="GROUP_2:th.177">Th177</a></i>;<br/>
<a NAME="E6:143"/><b>then </b>
( <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">I</font></span>)</span> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="group_2.html#E1:143"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="group_2.html#E2:143"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="group_2.html#E3:143"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<b>hence </b><a NAME="E7:143"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>


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