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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="E1:71"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> is <a href="tops_1.html#V1" target="_self">dense</a>
 ;<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><a NAME="E2:71"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>thus </b><a NAME="E3:71"/>
 <a href="pre_topc.html#K4">Cl</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"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b><i><font color="Green">E37</font></i>: 
not <font color="Maroon">TS</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>assume </b><a NAME="E2:71_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 ;<br/><a NAME="E3:71_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font>
 <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>, </i>;<br/><a NAME="E4:71_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</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="E5:71_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, </i>;<br/><a NAME="E6:71_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><a NAME="E7:71_1_1"/>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</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="E8:71_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <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/><b>hence </b><a NAME="E9:71_1_1"/>
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:71_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, </i>;<br/>


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

<a NAME="E4:71"/>
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</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="E5:71"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>


</div>
