<?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/>



<b>thus </b><a NAME="E1:55"/>
 <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><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="E1:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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>
 
;<br/>
<a NAME="E3:55_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">X</font> &amp; not <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><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:55_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<a NAME="E5:55_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:55_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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#K3">`</a> 
 
<b>by </b><i><a class="txt" href="roughs_1.html#E5:55_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:55_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:55_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="roughs_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/>


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

<b>thus </b><a NAME="E2:55"/>
 <a href="roughs_1.html#K5" target="_self">BndAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/>

<a NAME="E2:55_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:55_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:55_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:55_2"/><b>then </b><i><font color="Green">E35</font></i>: 
not <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><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:55_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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#K3">`</a> 
 
<b>by </b><i><a class="ref" href="roughs_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:55_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<a NAME="E8:55_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font>
 ;<br/>


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


</div>
