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

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <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="E1:66"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:66_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> &amp; <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E3:66_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> <font color="Maroon">X</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>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:66_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> <font color="Maroon">X</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:66_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">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:66_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:66"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_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="E1:66_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:66_2"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> &amp; <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E3:66_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <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> <font color="Maroon">X</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>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:66_2"/><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> <font color="Maroon">X</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:66_2"/><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">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E6:66_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="partfun1.html#E1:66_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="partfun1.html#E2:66_2"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="partfun1.html#E3:66_2"><i><font color="Green">E33</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="partfun1.html#E3:66_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="partfun1.html#E6:66_2"><i><font color="Green">E34</font></i></a>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
<span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="partfun1.html#K1" target="_self">*</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="partfun1.html#K3" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Z</font></span><a href="partfun1.html#K3" target="_self">:&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div>
