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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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>







<a NAME="E1:111"/>
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="group_2.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E2:111"/>
( <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font> implies <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T66">XBOOLE_1:66</a></i>;<br/>

<b>assume </b><a NAME="E3:111"/>
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:111"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E6:111"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:111"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T51" target="_self">Th51</a>, <a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E11:111"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, <a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/>
<font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="group_1.html#T21">GROUP_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<a NAME="E13:111"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T22">GROUP_1:22</a></i>;<br/>
<a NAME="E14:111"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:111"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E16:111"/>
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="group_2.html#T50" target="_self">Th50</a></i>;<br/>


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