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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="roughs_1.html#NM1" title="ROUGHS_1:NM.1">Approximation_Space</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>thus </b><a NAME="E1:57"/>
 <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="roughs_1.html#T14" target="_self" title="ROUGHS_1:th.14">Th14</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<a NAME="E3:57"/><b>then </b>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="roughs_1.html#T10" target="_self" title="ROUGHS_1:th.10">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:57"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E6:57"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="roughs_1.html#E5:57"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="roughs_1.html#T8" target="_self" title="ROUGHS_1:th.8">Th8</a></i>;<br/>
<a NAME="E7:57"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="roughs_1.html#E2:57"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="roughs_1.html#E5:57"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E8:57"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="roughs_1.html#E2:57"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="roughs_1.html#E7:57"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="roughs_1.html#E1:57_1"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="roughs_1.html#E2:57"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>


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