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<div class="add">

<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/>

<b>assume </b><a NAME="E2:71"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>thus </b><a NAME="E3:71"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</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="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
not <font color="Maroon" title="c1">TS</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="c5">Q1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/><b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c5">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><b>assume </b><a NAME="E2:71_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q1</font>
 ;<br/><a NAME="E3:71_1_1"/><b>then </b>
<font color="Maroon" title="c3">Q</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q1</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:71_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="tops_1.html#E2:71_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="tops_1.html#E1:71_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self" title="TOPS_1:th.39">Th39</a></i>;<br/><a NAME="E4:71_1_1"/><b>then </b><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:71_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_1.html#E2:71"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="tops_1.html#E1:71_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="tops_1.html#T38" target="_self" title="TOPS_1:th.38">Th38</a></i>;<br/><a NAME="E6:71_1_1"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/><a NAME="E7:71_1_1"/>
<font color="Maroon" title="c2">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:71"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="tops_1.html#E4:71_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="tops_1.html#E5:71_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="tops_1.html#T80" target="_self" title="TOPS_1:th.80">Th80</a></i>;<br/><a NAME="E8:71_1_1"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E9:71_1_1"/>
<font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q1</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E6:71_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:71_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:71_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="tops_1.html#E2:71_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self" title="TOPS_1:th.39">Th39</a></i>;<br/>


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

<a NAME="E4:71"/>
<font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:71"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P</font></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> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>


</div>
