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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="compts_1.html#V4" title="COMPTS_1:attr.4">being_T3</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="compact1.html#V5" title="COMPACT1:attr.5">locally-compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compact1.html#D3" target="_self" title="COMPACT1:def.3">COMPACT1:def 3</a><br/>

<b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> ) </span>}</span> ;<br/>
<a NAME="E2:13_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> ) </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> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_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="E1:13_1_1_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">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:13_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">Q</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> )
 
;<br/>
<b>hence </b><a NAME="E3:13_1_1_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> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:13_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c4">B</font> is    <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a><br/>

<b>consider </b><font color="Maroon" title="c6">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E2:13_1_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c6">W</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E1:13_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="compact1.html#D6" target="_self" title="COMPACT1:def.6">Def4</a></i>;<br/>
<a NAME="E3:13_1_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:13_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E5:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">W</font></span>)</span>
 
<b>by </b><i><a class="txt" href="compact1.html#E3:13_1_1_2"><i><font color="Green" title="E10">A5</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="E6:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c6">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T46" title="TOPS_1:th.46">TOPS_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:13_1_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c7">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E9:13_1_1_2"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c6">W</font>
 <b>and </b><br/><a NAME="E10:13_1_1_2"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T36" title="YELLOW_8:th.36">YELLOW_8:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Q</font> = <font color="Maroon" title="c7">Q</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> <b>by </b><i><a class="txt" href="compact1.html#E10:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Q</font>
;<br/>

<a NAME="E12:13_1_1_2"/>
<font color="Maroon" title="c5">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="compact1.html#E9:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="compact1.html#E2:13_1_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="compact1.html#E8:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="compts_1.html#T18" title="COMPTS_1:th.18">COMPTS_1:18</a></i>;<br/>
<b>hence </b><a NAME="E15:13_1_1_2"/>
<font color="Maroon" title="c8">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="compact1.html#E9:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>

<a NAME="E16:13_1_1_2"/>
<font color="Maroon" title="c5">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E17:13_1_1_2"/>
<font color="Maroon" title="c8">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="txt" href="compact1.html#E9:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:13_1_1"/>
<font color="Maroon" title="c4">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compact1.html#D1" target="_self" title="COMPACT1:def.1">COMPACT1:def 1</a><br/>

<b>assume </b><a NAME="E1:13_1_1_3"/>
<font color="Maroon" title="c5">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<a NAME="E2:13_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">Q</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Maroon" title="c5">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">W</font> )
 
;<br/>
<b>hence </b><a NAME="E3:13_1_1_3"/>
<font color="Maroon" title="c5">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E4:13_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div>
