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<b>let </b><font color="Maroon">T</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">T</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">A</font>
 ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font>;<br/><b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:16_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E3:16_1"/><b>then </b>
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/><b>hence </b><a NAME="E4:16_1"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">U</font> &amp; <font color="Maroon">U</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">B</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font><br/> for <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/>( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:16"/>
for <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> holds <br/>( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font>;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:16_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E5:16_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16_2"/>
<font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E7:16_2"/><b>then </b>
<font color="Maroon">U</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</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_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E8:16_2"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">U</font> &amp; <font color="Maroon">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T61">TOPS_1:61</a></i>;<br/>


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