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

<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<a NAME="E1:86"/>
for <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</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="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> iff <font color="Olive">q</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"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:86_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</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="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> implies <font color="Maroon">q</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"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:86_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:86_1_1"/><i><font color="Green">E37</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">f</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K15" target="_self">:]</a></span>
 <b>and </b><br/><a NAME="E4:86_1_1"/><i><font color="Green">E38</font></i>: 
<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"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:86_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">q</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">f</font>,<font color="Maroon">g</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><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:86_1_1"/>
 <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>
 
<b>by </b><i><a class="txt" href="funct_3.html#E9:86_1_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:86_1_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> )
 <b>and </b><br/><a NAME="E9:86_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">x</font>,<font color="Maroon">y</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="E10:86_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q</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">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>
 
<b>by </b><i><a class="txt" href="funct_3.html#E5:86_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="funct_3.html#E8:86_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E11:86_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="txt" href="funct_3.html#E8:86_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:86_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <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#E6:86_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="funct_3.html#E9:86_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="txt" href="funct_3.html#E9:86_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:86_1_1"/>
<font color="Maroon">q</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"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</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/>


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

<b>assume </b><a NAME="E2:86_1"/>
<font color="Maroon">q</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"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<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="E4:86_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:86_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:86_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</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#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:86_1"/><i><font color="Green">E52</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">f</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>and </b><br/><a NAME="E9:86_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:86_1"/><i><font color="Green">E107</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">g</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E12:86_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E13:86_1"/>
( <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> <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; <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> <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"><font color="Maroon">f</font>,<font color="Maroon">g</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></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">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">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <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">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</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="txt" href="funct_3.html#E6:86_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="funct_3.html#E9:86_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E14:86_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="funct_3.html#E5:86_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="funct_3.html#E8:86_1"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:86"/>
<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="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</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"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
