<?xml version="1.0"?>
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<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/>

<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<a NAME="E3:13"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span><a href="funct_3.html#K15">:]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E5:13"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E6:13"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="isocat_1.html#T1">ISOCAT_1:1</a></i>;<br/>
<a NAME="E8:13"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="isocat_1.html#T1">ISOCAT_1:1</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="funct_4.html#K2">~</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E10:13"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span></span><a href="funct_3.html#K15">:]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<i><font color="Green">E46</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T47">FUNCT_4:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span></span><a href="funct_3.html#K15">:]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span>
<b>by </b><i>, <a class="txt" href="functor0.html#E1:13"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T48">FUNCT_4:48</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:13_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E2:13_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E3:13_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T48">FUNCT_4:48</a></i>;<br/><a NAME="E4:13_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T88">FUNCT_3:88</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="E6:13_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:13_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E8:13_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">y</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#T103">ZFMISC_1:103</a></i>;<br/><b>set </b><font color="Maroon">x1</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>;<br/><b>set </b><font color="Maroon">x2</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font>;<br/><a NAME="E9:13_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">y2</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#K2">"</a> </span>)</span> &amp; <font color="Maroon">y1</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">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E10:13_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y1</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"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><i><font color="Green">E70</font></i>: <font color="Maroon">x</font> = 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span><a href="funct_3.html#K15">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y2</font>,<font color="Maroon">y1</font>
<b>by </b><i>, <a class="txt" href="functor0.html#E2:13"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor0.html#E3:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/><a NAME="E12:13_2_1"/>
( <font color="Maroon">y1</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">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">y2</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#K2">"</a> </span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E13:13_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:13_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E15:13_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span></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"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:13_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span></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">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T47">FUNCT_4:47</a></i>;<br/><b>thus </b><a NAME="E17:13_2_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">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="functor0.html#E4:13"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T47">FUNCT_4:47</a></i>;<br/><a NAME="E18:13_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor0.html#E5:13"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="functor0.html#E3:13"><i><font color="Green">E26</font></i></a>, , , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/></div>
<b>end;</b></div><b>assume </b><b>that </b><br/><a NAME="E2:13_2"/><i><font color="Green">E75</font></i>: 
<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">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:13_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><b>thus </b><a NAME="E4:13_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span>
 <b>by </b><i>, <a class="txt" href="functor0.html#E2:13"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:13_2"/>
<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#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T47">FUNCT_4:47</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:13_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E8:13_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E9:13_2"/><i><font color="Green">E79</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">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">y</font> = 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x1</font>
<b>by </b><i>, <a class="txt" href="functor0.html#E2:13"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor0.html#E3:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>
<br/>.= 
<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">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/><a NAME="E11:13_2"/>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:13_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:13_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T88">FUNCT_3:88</a></i>;<br/><a NAME="E14:13_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></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"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E15:13_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</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"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor0.html#E3:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:13"/>
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15">:]</a></span></span>)</span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a>  <a href="funct_4.html#K2">~</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K15">:]</a></span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>


</div>
