<?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/>



<a NAME="E1:53"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<b>assume </b><a NAME="E2:53"/>
<font color="Maroon">T</font> is <a href="relat_2.html#V8">transitive</a>
 ;<br/>

<a NAME="E3:53"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">T</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:53"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">N</font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font>;<br/>
<a NAME="E5:53"/>
for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</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="53_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:53_1"/><i><font color="Green">E29</font></i>: 
<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">T</font>
 
<b>by </b><i><a class="txt" href="toler_1.html#E4:53"><i><font color="Green">E27</font></i></a>, <a class="txt" href="toler_1.html#E1:53"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E3:53_1"/>
<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>
 
<b>by </b><i><a class="txt" href="toler_1.html#E4:53"><i><font color="Green">E27</font></i></a>, <a class="txt" href="toler_1.html#E1:53"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:53_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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#T27" target="_self">Th27</a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a>, <a class="txt" href="toler_1.html#E4:53"><i><font color="Green">E27</font></i></a>, <a class="txt" href="toler_1.html#E1:53_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Z</font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">T</font>,<font color="Maroon">x</font> 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/>
<a NAME="E7:53"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</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="53_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E36</font></i>: 
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<b>assume </b><a NAME="E2:53_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:53_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T32" target="_self">Th32</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E4:53_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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/>;<br/>
<a NAME="E5:53_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> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:53_2"/>
contradiction
 <b>by </b><i><a class="txt" href="toler_1.html#E4:53"><i><font color="Green">E27</font></i></a>, <a class="txt" href="toler_1.html#E1:53"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
 <a href="toler_1.html#NK4" target="_self">neighbourhood</a> , is   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
