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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">H1</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>







<b>thus </b><a NAME="E1:115"/>
<font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="group_2.html#K9" target="_self">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_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="E1:115_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="group_2.html#K9" target="_self">/\</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<b>then 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="E3:115_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:115_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font> <a href="group_2.html#K9" target="_self">/\</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E5:115_1"/>
( <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:115_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, <a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:115_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</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="E2:115"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<a NAME="E3:115"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then 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="E5:115"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:115"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:115"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g1</font>
 <b>and </b><br/><a NAME="E9:115"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, <a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E10:115"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , <a class="ref" href="group_1.html#T14">GROUP_1:14</a></i>;<br/>
<a NAME="E11:115"/><b>then </b>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font> <a href="group_2.html#K9" target="_self">/\</a> <font color="Maroon">H2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E12:115"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="group_2.html#K9" target="_self">/\</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/>


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