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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>









<b>assume </b><a NAME="E1:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><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><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:24"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:24_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:24_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <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="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:24_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:24_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:24_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:24_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:24_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
 <a href="relat_1.html#K1" target="_self">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<a NAME="E4:24"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:24_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_2_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:24_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <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="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:24_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:24_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:24_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:24_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:24_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:24_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:24_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:24"/>
 <a href="relat_1.html#K2" target="_self">rng</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
