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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>;<br/>



<b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> : verum </span>}</span> ;<br/>
<a NAME="E1:111"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</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="E1:111_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:111_1"/><b>then </b>
 ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font>
 
;<br/>
<b>hence </b><a NAME="E3:111_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:111"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> : verum </span>}</span>  is   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T45" target="_self" title="RLTOPSP1:th.45">Th45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:111"/>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E5:111"/><i><font color="Green" title="E71">A1</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T59" target="_self" title="RLTOPSP1:th.59">Th59</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">W</font>
;<br/>

<b>thus </b><a NAME="E6:111"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:111"><i><font color="Green" title="E71">A1</font></i></a></i>;<br/>


</div>
