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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Olive" title="b1">H</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) holds <br/> ex <font color="Olive" title="b2">C</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Olive" title="b1">H</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) implies  ex <font color="Olive" title="b1">C</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:147"/><i><font color="Green" title="E93">A1</font></i>: 
(  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> )</span><br/>

<i><font color="Green" title="E94">A2</font></i>:  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font></span>)</span> = 
 <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T161" target="_self" title="GROUP_3:th.161">Th161</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p2">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span>

;<br/>
<a NAME="E3:147"/><b>then </b><i><font color="Green" title="E95">A3</font></i>: 
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font>, <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<a NAME="E4:147"/><b>then </b>
(  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="group_3.html#E1:147"><i><font color="Green" title="E93">A1</font></i></a>, <a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:147"/><i><font color="Green" title="E96">A4</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">C</font> =  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="group_3.html#E1:147"><i><font color="Green" title="E93">A1</font></i></a>, <a class="txt" href="group_3.html#E3:147"><i><font color="Green" title="E95">A3</font></i></a>, <a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E8:147"/>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span></span><br/>

<b>thus </b><a NAME="E9:147"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E2:147"><i><font color="Green" title="E94">A2</font></i></a>, <a class="txt" href="group_3.html#E6:147"><i><font color="Green" title="E96">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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