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

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:82"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<b>thus </b><a NAME="E2:82"/>
 <a href="pscomp_1.html#K6">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
  <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="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:82_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:82_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:82_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:82_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">G</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:82_1_1"/>
<font color="Maroon">H</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T6">JORDAN5A:6</a></i>;<br/>
<a NAME="E6:82_1_1"/><b>then </b>
not <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#T38">PSCOMP_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1_1"/>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:82_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</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="E3:82"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E4:82"/>
for <font color="Olive">O</font> being <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/> not <font color="Olive">O</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">O</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:82_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i><a class="ref" href="jordan5a.html#T6">JORDAN5A:6</a></i>;<br/>
<a NAME="E4:82_2"/><b>then </b>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:82_2"/><b>then </b>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:82_2"/>
not <font color="Maroon">O</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:82"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T38">PSCOMP_1:38</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div>
