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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> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">a</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">X</font> be    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>;<br/>









<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">I</font> = <font color="Maroon" title="c4">X</font>;<br/>
<b>thus </b><a NAME="E2:4"/>
<font color="Maroon" title="c5">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green" title="E4">A2</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c8">g</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="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E5:4_1"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E6:4_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="group_6.html#E3:4_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:4"/>
<font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">b</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="E3:4_2"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="group_6.html#E3:4_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E6:4_2"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E7:4_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="group_6.html#E3:4_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:4"/>
<font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:4_3"/><i><font color="Green" title="E4">A4</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c8">g</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="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E5:4_3"/>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E6:4_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="group_6.html#E3:4_3"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:4"/>
<font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_4"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">b</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="E3:4_4"/><i><font color="Green" title="E4">A5</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c8">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="group_6.html#E3:4_4"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E6:4_4"/>
<font color="Maroon" title="c8">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E7:4_4"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="group_6.html#E3:4_4"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>


</div><b>end;</b></div>


</div>
