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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E2:64_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:64_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:64_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:64_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="txt" href="tex_2.html#E1:64_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1"><b>now </b></a><div class="add"><b>take </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="E1:64_1_1_1_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> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><b>hence </b><a NAME="E2:64_1_1_1_1"/>
<font color="Maroon">F</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="E3:64_1_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:64_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</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_1.html#T49">XBOOLE_1:49</a></i>;<br/><a NAME="E5:64_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>;<br/><a NAME="E6:64_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>;<br/><b>hence </b><a NAME="E7:64_1_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:64_1_1_1"/>
 ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:64_1_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:64_1_1_2_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:64_1_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E5:64_1_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1"><b>now </b></a><div class="add"><b>take </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/><b>thus </b><a NAME="E1:64_1_1_2_1_1"/>
<font color="Maroon">G</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="E2:64_1_1_2_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E3:64_1_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</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_1.html#T49">XBOOLE_1:49</a></i>;<br/><a NAME="E4:64_1_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>;<br/><a NAME="E5:64_1_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>;<br/><b>hence </b><a NAME="E6:64_1_1_2_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:64_1_1_2_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:64_1_1_2"/>
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>


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