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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">Q</font>, <font color="Maroon" title="c3">K</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="E1:19"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<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="E2:19"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> <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> <font color="Maroon" title="c3">K</font></span>)</span>
 ;<br/>

<a NAME="E3:19"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E17">A4</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">K</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p''</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TS</font> <b>by </b><i><a class="txt" href="tops_1.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E18">A5</font></i>: 
not <font color="Maroon" title="c1">TS</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E7:19"/>
for <font color="Olive" title="b1">Q1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">p''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q1</font> holds <br/><font color="Maroon" title="c2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:19_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c6">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:19_1"/>
<font color="Maroon" title="c5">p''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font>
 ;<br/>

<a NAME="E3:19_1"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c5">p''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E3:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</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#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="tops_1.html#E1:19_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tops_1.html#T38" target="_self" title="TOPS_1:th.38">Th38</a></i>;<br/>
<a NAME="E5:19_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">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="E6:19_1"/>
<font color="Maroon" title="c3">K</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E4:19"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="tops_1.html#E6:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="tops_1.html#E3:19_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="tops_1.html#E4:19_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self" title="TOPS_1:th.39">Th39</a></i>;<br/>
<a NAME="E7:19_1"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</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="E8:19_1"/>
<font color="Maroon" title="c2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">Q1</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E5:19_1"><i><font color="Green" title="E22">A9</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="E8:19"/>
<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="c2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="tops_1.html#E6:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self" title="TOPS_1:th.39">Th39</a></i>;<br/>


</div>
