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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">H1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">H</font>,<font color="Olive">H1</font> <a href="group_3.html#R5" target="_self">are_conjugated</a>  );<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="group_3.html#K1" target="_self">Subgroups</a> <font color="Maroon">G</font></span>)</span><b> such that </b><br/><a NAME="E2:112_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1" target="_self">Subgroups</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1">SUBSET_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E3:112_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> implies  ex <font color="Olive">H1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">H</font>,<font color="Olive">H1</font> <a href="group_3.html#R5" target="_self">are_conjugated</a>  ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E4:112_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">H</font>,<font color="Maroon">H1</font> <a href="group_3.html#R5" target="_self">are_conjugated</a>  )
 ;<br/>

<a NAME="E5:112_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1" target="_self">Subgroups</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T38" target="_self">Th38</a>, </i>;<br/>
<b>hence </b><a NAME="E6:112_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="group_3.html#T38" target="_self">Th38</a></i>;<br/>


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