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<div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>



<b>thus </b><a NAME="E1:63"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K6" target="_self">commutators</a> <font color="Maroon">G</font> implies  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <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="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K6" target="_self">commutators</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span>
 
;<br/>
<a NAME="E3:63_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <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="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span> &amp; <font color="Olive">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font> &amp; <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:63_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <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="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 ;<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>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:63"/><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>
 ;<br/>

<a NAME="E3:63"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K6" target="_self">commutators</a> <font color="Maroon">G</font>
 ;<br/>


</div>
