<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> is <a href="rltopsp1.html#V3" target="_self">circled</a>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:128"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">V</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E4:128"/>
<font color="Maroon">P</font> is   <a href="rltopsp1.html#NM2">local_base</a> of <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2">YELLOW13:def 2</a><br/>

<b>consider </b><font color="Maroon">W</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:128_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">W</font> is <a href="rltopsp1.html#V3" target="_self">circled</a>
 <b>and </b><br/><a NAME="E3:128_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E4:128_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/>

<b>thus </b><a NAME="E5:128_1"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D7" target="_self">RLTOPSP1:def 7</a><br/>

<b>assume </b><a NAME="E5:128"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>hence </b><a NAME="E6:128"/>
<font color="Maroon">V</font> is <a href="rltopsp1.html#V3" target="_self">circled</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a></i>;<br/>


</div>
