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

<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<a NAME="E1:70"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:70_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:70_1_1"/><b>then </b>
<span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E5:70_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:70_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E7:70_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:70_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:70_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:70_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:70_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E3:70_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:70_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:70_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:70_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:70_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70"/>
<span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
