<?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>;<br/>

<a NAME="E1:80"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <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>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <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>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:80"/><i><font color="Green">E38</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <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>)</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <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>)</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>, </i>;<br/>
<a NAME="E3:80"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></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> <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> &amp;  <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> <span class="p2">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></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> <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="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E4:80"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15" target="_self">:]</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> &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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="funct_3.html#K13" target="_self">:&gt;</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>, </i>;<br/>
<a NAME="E5:80"/>
for <font color="Olive">x</font>, <font color="Olive">y</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> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E42</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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:80_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></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#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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<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">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="funct_3.html#E2:80_1"><i><font color="Green">E43</font></i></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </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">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</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">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_3.html#T1" target="_self">Th1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:80"/>
<span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K13" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9" target="_self">pr1</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10" target="_self">pr2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="funct_3.html#K13" target="_self">:&gt;</a></span>
 <b>by </b><i>, </i>;<br/>


</div>
