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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b3">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">C</font> = <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">K</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E1:109"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">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="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E2:109"/><span class="kw">then </span>
 not  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">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="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">P</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:109"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c6">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T57" target="_self" title="RLTOPSP1:th.57">Th58</a>, <a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E5:109"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">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">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">P9</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">P</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T3" title="CONNSP_2:th.3">CONNSP_2:3</a></span>;<br/>
<a NAME="E7:109"/>
( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c6">P</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c6">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rltopsp1.html#E13"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<a NAME="E8:109"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a></span>;<br/>
<a NAME="E9:109"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T36" title="TSEP_1:th.36">TSEP_1:36</a></span>;<br/>
<a NAME="E10:109"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">P9</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="rltopsp1.html#T12" target="_self" title="RLTOPSP1:th.12">Th12</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E11:109"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T35" title="CONVEX1:th.35">CONVEX1:35</a></span>;<br/>
<a NAME="E12:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A3</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T24" title="SUBSET_1:th.24">SUBSET_1:24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E14:109"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
<font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E15:109"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">W</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E16:109"/>
<font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:109"><span class="lab"><font color="Green" title="E72">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font></span><br/>

<a NAME="E17:109"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:109"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="pre_topc.html#T19" title="PRE_TOPC:th.19">PRE_TOPC:19</a></span>;<br/>
<a NAME="E18:109"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<a NAME="E19:109"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:109"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:109"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:109"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
