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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<b>thus </b><a NAME="E1:120"/>
( <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="group_3.html#R5" target="_self">are_conjugated</a>  implies  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>, <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> <a href="group_3.html#R4" target="_self">are_conjugated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E1:120_1"/><i><font color="Green">E38</font></i>: 
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H1</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_3.html#D11" target="_self">GROUP_3:def 11</a><br/>

<a NAME="E2:120_1"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, <a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:120_1"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>, <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> <a href="group_3.html#R4" target="_self">are_conjugated</a> 
 <b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:120"/><i><font color="Green">E39</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_3.html#D9" target="_self">GROUP_3:def 9</a><br/>

<a NAME="E3:120"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, <a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E4:120"/><b>then </b>
<font color="Maroon">H1</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H2</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T68">GROUP_2:68</a></i>;<br/>
<b>hence </b><a NAME="E5:120"/>
<font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="group_3.html#R5" target="_self">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="group_3.html#E37"><i><font color="Green">Lemma74</font></i></a></i>;<br/>


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