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

<b>consider </b><font color="Maroon">A0</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>;<br/>
<a NAME="E2:4_1_1"/>
<font color="Maroon">A0</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E3:4_1_1"/>
not  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>;<br/>
<a NAME="E4:4_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>
<a NAME="E5:4_1_1"/><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E6:4_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is  not    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E7:4_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E8:4_1_1"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:4_1_1"/>
 <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i>, , <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>

<b>thus </b><a NAME="E10:4_1_1"/>
 <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> is <a href="waybel_0.html#V13">upper</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a><br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>
<b>assume </b><a NAME="E4:4_1_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E5:4_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E6:4_1_1_1"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:4_1_1_1"/><b>then </b>
<font color="Maroon">b</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:4_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 ;<br/>


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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E11:4_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i/>;<br/>
<a NAME="E13:4_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T4">CONNSP_2:4</a></i>;<br/>
<a NAME="E14:4_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E16:4_1_1"/>
( <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E17:4_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>


</div>
