<?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> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:19"/>
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R2">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:19_1"/>
<span class="p1"><a href="tarski.html#K1">{</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>4</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <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>4</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>4</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <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>4</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>4</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="pzfmisc1.html#E1:19_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E1:19_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</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="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:19"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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>4</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <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>4</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>4</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<a NAME="E4:19"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <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>4</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>4</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="pzfmisc1.html#E2:19"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:19"/>
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E2:19"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D2" target="_self">Def2</a></i>;<br/>


</div>
