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<b>let </b><font color="Maroon" title="c1">GX</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 ;<br/><a NAME="E2:79_1"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4" target="_self" title="TOPS_1:def.4">Def4</a></i>;<br/><a NAME="E3:79_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#D3" target="_self" title="TOPS_1:def.3">Def3</a></i>;<br/><a NAME="E4:79_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E5:79_1"/>
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:79"/>
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">R</font>
 ;<br/>

<a NAME="E3:79"/><b>then </b>
( <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:79"/><b>then </b><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:79"/>
<font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:79"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E4:79"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E7:79"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E8:79"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E9:79"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3" target="_self" title="TOPS_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E10:79"/>
<font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4" target="_self" title="TOPS_1:def.4">Def4</a></i>;<br/>


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