<?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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/>
not <font color="Maroon">X</font> is <a href="roughs_1.html#V4" target="_self">rough</a>
 ;<br/><a NAME="E2:39_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="E3:39_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="E4:39_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <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="E5:39_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/><a NAME="E6:39_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <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>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:39_1"/>
<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/><b>hence </b><a NAME="E8:39_1"/>
 <a href="roughs_1.html#K4" target="_self">UAp</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="E2:39"/><i><font color="Green">E33</font></i>: 
 <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:39"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:39_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:39_2"/>
 <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">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><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="E1:39_2_1"/><i><font color="Green">E35</font></i>: 
<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">x</font>
 ;<br/>

<a NAME="E2:39_2_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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E3:39_2_1"/><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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<a NAME="E4:39_2_1"/><b>then </b>
<font color="Maroon">x</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>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E5:39_2_1"/><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>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E6:39_2_1"/>
<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>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon">x</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/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:39"/><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="E5:39"/><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="E6:39"/>
not <font color="Maroon">X</font> is <a href="roughs_1.html#V4" target="_self">rough</a>
 <b>by </b><i/>;<br/>


</div>
