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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>







<a NAME="E1:105"/>
<font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="group_2.html#T130" target="_self" title="GROUP_2:th.130">Th130</a></i>;<br/>
<b>hence </b><a NAME="E2:105"/>
( <font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> implies <font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T66" title="XBOOLE_1:th.66">XBOOLE_1:66</a></i>;<br/>

<b>assume </b><a NAME="E3:105"/>
<font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:105"/><i><font color="Green" title="E76">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font>
 <b>and </b><br/><a NAME="E6:105"/><i><font color="Green" title="E77">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_2.html#E6:105"><i><font color="Green" title="E77">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:105"/><i><font color="Green" title="E78">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c7">g</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">H</font> )
 <b>by </b><i><a class="txt" href="group_2.html#E5:105"><i><font color="Green" title="E76">A1</font></i></a>, <a class="ref" href="group_2.html#T125" target="_self" title="GROUP_2:th.125">Th125</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E11:105"/><i><font color="Green" title="E79">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">H</font> )
 <b>by </b><i><a class="txt" href="group_2.html#E6:105"><i><font color="Green" title="E77">A2</font></i></a>, <a class="ref" href="group_2.html#T125" target="_self" title="GROUP_2:th.125">Th125</a></i>;<br/>
<font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_2.html#E9:105"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="group_2.html#E11:105"><i><font color="Green" title="E79">A4</font></i></a>, <a class="ref" href="group_1.html#T22" title="GROUP_1:th.22">GROUP_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/>
<a NAME="E13:105"/><b>then </b><i><font color="Green" title="E80">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T21" title="GROUP_1:th.21">GROUP_1:21</a></i>;<br/>
<a NAME="E14:105"/>
<font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E9:105"><i><font color="Green" title="E78">A3</font></i></a>, <a class="ref" href="group_2.html#T60" target="_self" title="GROUP_2:th.60">Th60</a></i>;<br/>
<a NAME="E15:105"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E11:105"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="group_2.html#E13:105"><i><font color="Green" title="E80">A5</font></i></a>, <a class="ref" href="group_2.html#T59" target="_self" title="GROUP_2:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E16:105"/>
<font color="Maroon" title="c2">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T137" target="_self" title="GROUP_2:th.137">Th137</a></i>;<br/>


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