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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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</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="E1:123"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>then 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="E3:123"/><i><font color="Green">E85</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>3</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>3</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="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E4:123"/><i><font color="Green">E86</font></i>: 
(  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  &amp;  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="group_2.html#E3:123"><i><font color="Green">E85</font></i></a>, <a class="ref" href="group_2.html#T154" target="_self">Th154</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>7</sub></font> =  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</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="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E6:123"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E3:123"><i><font color="Green">E85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:123"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E4:123"><i><font color="Green">E86</font></i></a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>


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