<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:88"/>
<span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></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="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:88_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E4:88_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#R2" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="group_3.html#T96" target="_self">Th96</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:88_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_3.html#E6:88_1"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T1" target="_self">Th1</a></i>
;<br/>
<b>then </b><i><font color="Green">E67</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="group_3.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T25" target="_self">Th25</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T29" target="_self">Th29</a></i>
;<br/>
<a NAME="E10:88_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="group_3.html#T97" target="_self">Th97</a></i>;<br/>
<b>hence </b><a NAME="E11:88_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_3.html#E9:88_1"><i><font color="Green">E67</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></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="E2:88"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:88"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E5:88"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#R2" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green">E65</font></i></a>, <a class="ref" href="group_3.html#T96" target="_self">Th96</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:88"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green">E65</font></i></a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_3.html#E7:88"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_3.html#T1" target="_self">Th1</a></i>
;<br/>
<b>then </b><i><font color="Green">E67</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T29" target="_self">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_3.html#T26" target="_self">Th26</a></i>
;<br/>
<a NAME="E11:88"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="group_3.html#T97" target="_self">Th97</a></i>;<br/>
<b>hence </b><a NAME="E12:88"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_3.html#E10:88"><i><font color="Green">E67</font></i></a></i>;<br/>


</div>
