<?xml version="1.0"?>
<div class="add">

<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>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="E1:78"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 <b>and </b><br/><a NAME="E2:78"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 <b>and </b><br/><a NAME="E3:78"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E4:78"/><i><font color="Green">E37</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#T45" target="_self">Th45</a>, <a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E5:78"/>
<font color="Maroon">Q</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#T46" target="_self">Th46</a>, <a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E6:78"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E7:78"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</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#K7">\</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, <a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E8:78"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</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#K7">\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</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#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E9:78"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</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#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</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#K7">\</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E10:78"/>
 <a href="pre_topc.html#K4">Cl</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#K7">\</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</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> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E11:78"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, <a class="ref" href="tops_1.html#T53" target="_self">Th53</a>, , <a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E12:78"/><b>then </b>
<font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E13:78"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E14:78"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, </i>;<br/>
<a NAME="E15:78"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T54" target="_self">Th54</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:78"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <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#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E17:78"/>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</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/>


</div>
