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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>;<br/>







<b>thus </b><a NAME="E1:13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span>
  <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="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="E1:13_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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="E3:13_1"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>
<a NAME="E4:13_1"/>
( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E3:13_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:13_1"/><b>then </b>
( <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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font> or <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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E3:13_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">x</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="E2:13"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1"/>
( <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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font> or <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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font> )
 </span><b>by </b><i><a class="txt" href="group_2.html#E2:13"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_1"/>
<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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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="E3:13_2_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/><a NAME="E4:13_2_1_1"/>
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:13_2_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:13_2_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_2"/>
<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#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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="E3:13_2_1_2"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/><a NAME="E4:13_2_1_2"/>
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:13_2_1_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:13_2_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:13_2_1_2"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c4">C</font>
 ;<br/>


</div>
