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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:57"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E4:57"/>
( <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> implies ( not <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E5</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:57_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pboole.html#D9">PBOOLE:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:57_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E3:57_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T72">ZFMISC_1:72</a></i>;<br/>
<b>hence </b><a NAME="E4:57_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:57_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


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

<b>assume </b><a NAME="E3:57_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pboole.html#D9">PBOOLE:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:57_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E5:57_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T72">ZFMISC_1:72</a></i>;<br/>
<b>hence </b><a NAME="E6:57_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E3:57"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E3:57_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


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


</div>
