<?xml version="1.0"?>
<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">T</font> be   <a href="eqrel_1.html#NM1">Tolerance</a> of <font color="Maroon">X</font>;<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>assume </b><a NAME="E1:35"/><i><font color="Green">E21</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> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:35"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1"/>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a>, , , <a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">PS</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> as    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="toler_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">Z</font> being   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:35"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">PS</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<a NAME="E7:35"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</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/>
<b>hence </b><a NAME="E8:35"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E1:35"><i><font color="Green">E21</font></i></a></i>;<br/>

<a NAME="E9:35"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</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/>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E1:35"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
