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

<b>let </b><font color="Maroon">A</font> be   <a href="roughs_1.html#NM2">Tolerance_Space</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>



<a NAME="E1:38"/><i><font color="Green">E24</font></i>: 
 <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1"/>
not <font color="Maroon">X</font> is <a href="roughs_1.html#V4" target="_self">rough</a>
 ;<br/><a NAME="E2:38_1"/><b>then </b>
 <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E3:38_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E4:38_1"/><b>then </b>
 <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E5:38_1"/><b>then </b>
 <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E6:38_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E7:38_1"/>
 <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:38_1"/>
 <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:38"/><i><font color="Green">E33</font></i>: 
 <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:38"/>
 <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:38_2"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>,<font color="Maroon">y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:38_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:38_2"/><i><font color="Green">E35</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>,<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:38_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:38_2"/>
<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="roughs_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:38"/><b>then </b>
<span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E6:38"/><b>then </b>
 <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>hence </b><a NAME="E7:38"/>
not <font color="Maroon">X</font> is <a href="roughs_1.html#V4" target="_self">rough</a>
 <b>by </b><i/>;<br/>


</div>
