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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E28">Th52</font></span>: <a NAME="T52"><span class="comment"><font color="firebrick">:: GROUP_5:52</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b3">H1</font>, <font color="Olive" title="b4">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b2">G</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> (<font color="Olive" title="b3">H1</font>,<font color="Olive" title="b4">H2</font>) iff  ex <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">G</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> &amp; <font color="Olive" title="b5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">H1</font> &amp; <font color="Olive" title="b6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b4">H2</font> ) )</div></div>
