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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:141"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T164" target="_self">Th164</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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">E97</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E1:141"><i><font color="Green">E96</font></i></a>, <a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E5:141"/><i><font color="Green">E98</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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E4:141"><i><font color="Green">E97</font></i></a>, <a class="ref" href="group_2.html#T167" target="_self">Th167</a></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:141_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:141_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E1:141_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="group_2.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E5:141_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="group_2.html#E1:141"><i><font color="Green">E96</font></i></a>, <a class="ref" href="group_2.html#T3" target="_self">Th3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E7:141_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E8:141_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="txt" href="group_2.html#E1:141"><i><font color="Green">E96</font></i></a>, <a class="ref" href="group_2.html#T48" target="_self">Th48</a></i>;<br/><a NAME="E9:141_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T156" target="_self">Th156</a></i>;<br/><b>hence </b><a NAME="E10:141_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E4:141_1"><i><font color="Green">E100</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></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">E101</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E12:141_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_2.html#E11:141_1"><i><font color="Green">E101</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:141_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E11:141_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="group_2.html#D15" target="_self">Def15</a></i>;<br/><b>thus </b><a NAME="E16:141_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="group_2.html#E4:141_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="group_2.html#E15:141_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="group_2.html#T151" target="_self">Th151</a></i>;<br/><b>thus </b><a NAME="E17:141_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E4:141_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="group_2.html#E12:141_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="group_2.html#E15:141_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="group_2.html#T138" target="_self">Th138</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:141"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></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<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_2.html#E199"><i><font color="Green">Lemma94</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:141"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E4:141"><i><font color="Green">E97</font></i></a>, <a class="txt" href="group_2.html#E5:141"><i><font color="Green">E98</font></i></a>, <a class="ref" href="group_2.html#D18" target="_self">Def18</a></i>;<br/>


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