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

<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>, <font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">k</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E3:67"/><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="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span>
 ;<br/>

<a NAME="E4:67"/><i><font color="Green">E42</font></i>: 
(  <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> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <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">k</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T4" target="_self">Th4</a>, <a class="txt" href="funct_3.html#E4:67"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E5:67"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_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>assume </b><a NAME="E1:67_1"/>
<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>
 ;<br/>

<a NAME="E2:67_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#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> &amp; <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">h</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"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</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="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_3.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:67"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_3.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<a NAME="E7:67"/><i><font color="Green">E43</font></i>: 
(  <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> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp;  <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">k</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T4" target="_self">Th4</a>, <a class="txt" href="funct_3.html#E4:67"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E8:67"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><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>assume </b><a NAME="E1:67_2"/>
<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>
 ;<br/>

<a NAME="E2:67_2"/><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#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> &amp; <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">h</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"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</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="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="funct_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:67_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:67"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="funct_3.html#E7:67"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
