<?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="E1:54"/>
 <a href="roughs_1.html#K4" target="_self">UAp</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="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_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="E1:54_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</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="E2:54_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>  <a href="roughs_1.html#K3" target="_self">LAp</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="E4:54_1"/><i><font color="Green">E24</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="E5:54_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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="txt" href="roughs_1.html#E7:54_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E6:54_1"/><b>then </b>
<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> 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="E7:54_1"/><b>then </b><i><font color="Green">E27</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">z</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E8:54_1"/>
 <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><a class="txt" href="roughs_1.html#E7:54_1"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:54_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#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:54"/>
 <a href="roughs_1.html#K3" target="_self">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span></span>)</span> <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/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:54"/>
 <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#K3" target="_self">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_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="E1:54_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#K3" target="_self">LAp</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:54_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>  <a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_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="E1:54_2_1"/><i><font color="Green">E34</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="E2:54_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="E3:54_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="E4:54_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="E5:54_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>  <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/>
<b>hence </b><a NAME="E6:54_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</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="roughs_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_2"/>
<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"><a href="roughs_1.html#K4" target="_self">UAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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


</div>
