<?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/>

<a NAME="E1:85"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</span> iff <font color="Olive">p</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#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:85_1"/><i><font color="Green">E37</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">h</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">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:85_1"/><i><font color="Green">E38</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">g</font>,<font color="Maroon">k</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">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:85_1"/>
( <font color="Maroon">p</font> <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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</span> implies <font color="Maroon">p</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#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_1_1"/>
<font color="Maroon">p</font> <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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:85_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p</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" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> &amp; <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</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="E4:85_1_1"/><i><font color="Green">E42</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> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> )
 <b>and </b><br/><a NAME="E5:85_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</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#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:85_1_1"/>
<span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:85_1_1"/><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">k</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"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="funct_3.html#E2:85"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="funct_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:85_1_1"/><b>then </b>
( <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#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</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#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:85_1_1"/><b>then </b>
( <font color="Maroon">x1</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#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">x2</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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E10:85_1_1"/>
<font color="Maroon">p</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#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></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/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:85_1"/>
<font color="Maroon">p</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#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<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="E6:85_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</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#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">x2</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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span> )
 <b>and </b><br/><a NAME="E7:85_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</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#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E8:85_1"/>
( <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#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</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#K1">dom</a> <font color="Maroon">h</font> &amp; <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> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="txt" href="funct_3.html#E2:85"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E9:85_1"/><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">k</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"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> &amp; <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" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:85_1"/><b>then </b>
( <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> &amp; <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" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E11:85_1"/>
<font color="Maroon">p</font> <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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:85"/><b>then </b><i><font color="Green">E50</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" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></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> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <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#NK3" target="_self">*</a> <font color="Maroon">g</font></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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span><a href="funct_3.html#K15" target="_self">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E3:85"/>
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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Olive">y</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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</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#K15" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:85_2"/><i><font color="Green">E51</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"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</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">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E3:85_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">g</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">f</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, <a class="txt" href="funct_3.html#E2:85"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E4:85_2"/><i><font color="Green">E107</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">g</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">k</font> )
 
<b>by </b><i>, <a class="txt" href="funct_3.html#E2:85"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E5:85_2"/><b>then </b>
( <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">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <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">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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#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">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="funct_3.html#E1:85_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:85_2"/><b>then </b>
( <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">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</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> <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">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E7:85_2"/><b>then </b>
( <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>  <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">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> &amp; <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <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>  <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">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</font></span><a href="funct_3.html#K15" target="_self">:]</a></span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</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>)</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#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="funct_3.html#E2:85_2"><i><font color="Green">E52</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"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="funct_3.html#E1:85_2"><i><font color="Green">E51</font></i></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> <font color="Maroon">g</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</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> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span> <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#E2:85"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></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>
<b>by </b><i>, <a class="txt" href="funct_3.html#E2:85"><i><font color="Green">E50</font></i></a>, </i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:85"/>
<span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="funct_3.html#K15" target="_self">:]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_3.html#K15" target="_self">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">k</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#K15" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">k</font></span>)</span></span><a href="funct_3.html#K15" target="_self">:]</a></span>
 <b>by </b><i>, </i>;<br/>


</div>
