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



<b>assume </b><a NAME="E1:80"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 ;<br/><a NAME="E2:80_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1" target="_self">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:80_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E5:80_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, </i>;<br/><a NAME="E6:80_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</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_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:80_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">TS</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/><a NAME="E9:80_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E10:80_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, </i>;<br/><a NAME="E11:80_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, </i>;<br/><a NAME="E12:80_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E13:80_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">G</font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T53" target="_self">Th53</a>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:80"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>  st <font color="Olive">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Q</font> &amp; <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">G</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:80_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:80_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E4:80_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E5:80_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:80_2"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:80_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a></i>;<br/><a NAME="E8:80_2"/>
<font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="tops_1.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E9:80_2"/><b>then </b>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</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="E10:80_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, <a class="ref" href="xboole_1.html#T67">XBOOLE_1:67</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:80"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1" target="_self">dense</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:80"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/>


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