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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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 ;<br/><a NAME="E2:79_1"/><b>then </b><i><font color="Green">E21</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="E3:79_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:79_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:79_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E6:79_1"/>
<font color="Maroon">Q</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</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/><b>hence </b><a NAME="E7:79_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:79"/><i><font color="Green">E38</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="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E3:79"/>
not <font color="Maroon">P</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 ;<br/>

<a NAME="E4:79"/><b>then </b>
not <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>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E6:79"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E7:79"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:79"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:79"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<b>hence </b><a NAME="E10:79"/>
contradiction
 <b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a></i>;<br/>


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