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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">H</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<b>thus </b><a NAME="E1:59"/>
 <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">H</font>,<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">H</font>,<font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:59_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E4:59_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:59_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:59_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E7:59_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> <a href="group_3.html#K6">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/>
<a NAME="E8:59_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="group_3.html#D13">GROUP_3:def 13</a></i>;<br/>
<a NAME="E9:59_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E10:59_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:59"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">N</font>,<font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:59"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E6:59"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:59"/>
<font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> <a href="group_3.html#K6">|^</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/>
<a NAME="E8:59"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , , <a class="ref" href="group_2.html#T60">GROUP_2:60</a>, <a class="ref" href="group_3.html#D13">GROUP_3:def 13</a></i>;<br/>
<a NAME="E9:59"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E10:59"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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