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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Olive" title="b1">H</font></span>)</span> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K16" title="GROUP_2:func.16">Right_Cosets</a> <font color="Olive" title="b1">H</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K16" title="GROUP_2:func.16">Right_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> )</span><br/>



<span class="kw">thus </span><a NAME="E1:170"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K16" title="GROUP_2:func.16">Right_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span></span><br/>

<a NAME="E2:170"/>
 <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c2">H</font>, <a href="group_2.html#K16" title="GROUP_2:func.16">Right_Cosets</a> <font color="Maroon" title="c2">H</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T166" target="_self" title="GROUP_2:th.166">Th166</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:170"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">Index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K16" title="GROUP_2:func.16">Right_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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