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<b>let </b><font color="Maroon">X</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">r3</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 ;<br/><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><b>that </b><br/><a NAME="E2:58_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E3:58_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">O</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E4:58_1"/>
<font color="Maroon">O</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:58_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E6:58_1"/>
<font color="Maroon">O</font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/><a NAME="E7:58_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E2:58_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E8:58_1"/>
<font color="Maroon">O</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><b>hence </b><a NAME="E9:58_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:58_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pscomp_1.html#E2:58"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:58"/><i><font color="Green">E36</font></i>: 
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">r3</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">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E3:58"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E4:58"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:58"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:58"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:58"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="subset_1.html#T40">SUBSET_1:40</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
not <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="txt" href="pscomp_1.html#E2:58"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:58"/>
<font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


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