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

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </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="E1:54_1"/>
<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"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E36</font></i>: 
<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">z2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:54_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font>;<br/><b>take </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font>;<br/><b>thus </b><a NAME="E5:54_1"/>
( <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">z2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="funct_4.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="funct_4.html#E1:54"><i><font color="Green">E34</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:54"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><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="E1:54_2"/>
<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"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:54_2"/><b>then </b>
 ex <font color="Olive">y2</font>, <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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="Olive">z2</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y2</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z2</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:54_2"/>
<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/>


</div><b>end;</b></div>
<a NAME="E3:54"/>
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> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><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="E1:54_3"/>
<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"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_3"/><i><font color="Green">E42</font></i>: 
<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">z2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:54_3"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E5:54_3"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="funct_4.html#E1:54"><i><font color="Green">E34</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="funct_4.html#E1:54"><i><font color="Green">E34</font></i></a>, , </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
 <a href="funct_4.html#K2" target="_self">~</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" target="_self">~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="funct_4.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div>
