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

<b>let </b><font color="Maroon">IT</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">ClX</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">ClIT</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:51_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">W</font> =  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:51_1_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E4:51_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E5:51_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T1">FCONT_3:1</a></i>;<br/>
<a NAME="E6:51_1_2"/>
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E7:51_1_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E8:51_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T1">FCONT_3:1</a></i>;<br/>
<b>thus </b><a NAME="E9:51_1_2"/>
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</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="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:51_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E2:51_1_2_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span>  holds <br/><font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:51_1_2_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:51_1_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:51_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>
<a NAME="E6:51_1_2_1_1"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span>  holds <br/><font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:51_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> is <a href="rcomp_1.html#V2">closed</a> )
 
;<br/>
<b>hence </b><a NAME="E3:51_1_2_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:51_1_2_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">IT</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E5:51_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pscomp_1.html#E8:51_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<a NAME="E8:51_1_2_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E4:51_1_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:51_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:51_1_2_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:51_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pscomp_1.html#E1:51_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pscomp_1.html#E3:51_1_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">e</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="E10:51_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 ;<br/>

<a NAME="E11:51_1_2"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span>  holds <br/><font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:51_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:51_1_2_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:51_1_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>
<a NAME="E6:51_1_2_2"/>
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E5:51_1_2"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E7:51_1_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : ( <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:51_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:51_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:51_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pscomp_1.html#E8:51_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div>
