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<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>assume </b><a NAME="E2:46_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><a NAME="E3:46_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">G1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>and </b><br/><a NAME="E6:46_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>and </b><br/><a NAME="E7:46_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:46_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">G2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E9:46_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">G2</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>and </b><br/><a NAME="E10:46_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E11:46_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:46_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pre_topc.html#E5:46_1"><i><font color="Green">E21</font></i></a>, </i>;<br/><a NAME="E12:46_1"/>
( <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font> &amp; <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E13:46_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/><a NAME="E14:46_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i/>;<br/><a NAME="E15:46_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:46_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="pre_topc.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="pre_topc.html#K1" target="_self">{}</a> <font color="Maroon">GX</font>

;<br/><a NAME="E18:46_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E19:46_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:46_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pre_topc.html#T22" target="_self">Th22</a>, <a class="ref" href="pre_topc.html#T23" target="_self">Th23</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:46"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:46"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:46"/><b>then </b>
(  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:46"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:46"/>
 <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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