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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E2:12_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><b>assume </b><a NAME="E3:12_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>assume </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E5:12_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
not <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E6:12_1_1_1"/>
not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E7:12_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:12_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E10:12_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font>;<br/><i><font color="Green">E25</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 

<br/>.= 
<font color="Maroon">F</font>

;<br/><a NAME="E12:12_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E13:12_1_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E14:12_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E15:12_1_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><b>hence </b><a NAME="E16:12_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:12_1_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>assume </b><a NAME="E2:12_1_1_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>assume </b><a NAME="E3:12_1_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>assume </b><a NAME="E4:12_1_1_2_1"/>
not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E5:12_1_1_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:12_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:12_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>;<br/><a NAME="E9:12_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E10:12_1_1_2_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><a NAME="E11:12_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E12:12_1_1_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E13:12_1_1_2_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</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="E14:12_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_1_2"/>
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>


</div>
