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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:19"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a> implies for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E13</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_3.html#D2" target="_self">TOPS_3:def 2</a><br/>

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:19_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E3:19_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:19"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E4:19"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:19"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T3" target="_self">Th3</a></i>;<br/>


</div>
