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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>







<b>thus </b><a NAME="E1:31"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<b>by </b><i><a class="ref" href="relset_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:31_1"/>
<span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:31_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:31_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E5:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:31_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:31_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="relset_2.html#E3:31_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="relset_2.html#E4:31_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:31_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>assume </b><a NAME="E6:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E7:31_1"/><b>then </b>
<span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <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="relset_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1"/>
contradiction
 <b>by </b><i><a class="txt" href="relset_2.html#E3:31_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="relset_2.html#E5:31_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


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

<b>assume </b><a NAME="E2:31"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:31"/><b>then </b>
<span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relset_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:31"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>


</div>
