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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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:77"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 ;<br/>

<a NAME="E3:77"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:77"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">G</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">G</font> holds <br/><font color="Maroon">S</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:77_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:77_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:77_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:77"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">G</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i><a class="txt" href="waybel_9.html#E3:77"><i><font color="Green">E34</font></i></a>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E5:77_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:77_1"/>
<font color="Maroon">S</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#T26">YELLOW_6:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:77"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>


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