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

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

<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b2">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Olive" title="b1">N</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )
 ;<br/><b>set </b><font color="Maroon" title="c3">G</font> =  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1;<br/><div><i><font color="Green" title="E28">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c5">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">x</font>;<br/><a NAME="E5:40_1_1"/>
( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E6:40_1_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">Y</font> &amp;  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="yellow19.html#E1:40_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="yellow19.html#E3:40_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><a NAME="E7:40_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 ;<br/><b>hence </b><a NAME="E8:40_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="yellow19.html#E3:40_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E29">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>and </b><br/><a NAME="E5:40_1"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="yellow19.html#E2:40_1"><i><font color="Green" title="E28">A3</font></i></a>);<br/></i><a NAME="E6:40_1"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E1:40_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="cantor_1.html#T4" title="CANTOR_1:th.4">CANTOR_1:4</a>, <a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c6">R</font> = <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> ;<br/><a NAME="E8:40_1"/><i><font color="Green" title="E32">A8</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c5">G</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c5">G</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></i>;<br/><a NAME="E9:40_1"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T12" title="YELLOW_6:th.12">YELLOW_6:12</a></i>;<br/><a NAME="E10:40_1"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a>,<a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a><br/>

<b>assume </b><a NAME="E1:40_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )
 ;<br/>

<a NAME="E2:40_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="yellow19.html#E9:40_1"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">X</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:40_1_2"/><i><font color="Green" title="E34">A10</font></i>: 
( <font color="Maroon" title="c9">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c9">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">X</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>
<a NAME="E5:40_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="yellow19.html#E9:40_1"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E7:40_1_2"/><i><font color="Green" title="E35">A11</font></i>: 
( <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c10">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">z</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font></span>)</span>;<br/>
<a NAME="E8:40_1_2"/>
( <font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="yellow19.html#E4:40_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="yellow19.html#E7:40_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">z</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <b>by </b><i><a class="txt" href="yellow19.html#E9:40_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">z</font>
;<br/>

<a NAME="E10:40_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 
;<br/>
<b>hence </b><a NAME="E11:40_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>

<a NAME="E12:40_1_2"/>
( <font color="Maroon" title="c9">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E13:40_1_2"/><b>then </b>
( <font color="Maroon" title="c12">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c12">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">y</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> )
 
<b>by </b><i><a class="txt" href="yellow19.html#E4:40_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="yellow19.html#E7:40_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a>, <a class="ref" href="setfam_1.html#T59" title="SETFAM_1:th.59">SETFAM_1:59</a></i>;<br/>
<a NAME="E14:40_1_2"/><b>then </b>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c12">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c12">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">y</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E15:40_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> )
 <b>by </b><i><a class="ref" href="yellow_7.html#T1" title="YELLOW_7:th.1">YELLOW_7:1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">R</font> = <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow19.html#E4:40_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="yellow19.html#E9:40_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>set </b><font color="Maroon" title="c9">N</font> = <font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font>;<br/><a NAME="E13:40_1"/><i><font color="Green" title="E34">A12</font></i>: 
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">R</font> #) &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">f</font> )
 <b>by </b><i><a class="ref" href="waybel32.html#D3" title="WAYBEL32:def.3">WAYBEL32:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c10">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E14:40_1"/><i><font color="Green" title="E35">A13</font></i>: 
 <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D3" title="YELLOW_6:def.3">YELLOW_6:def 3</a></i>;<br/><a NAME="E15:40_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> )
 <b>by </b><i><a class="ref" href="classes1.html#T5" title="CLASSES1:th.5">CLASSES1:5</a>, <a class="ref" href="classes1.html#T59" title="CLASSES1:th.59">CLASSES1:59</a></i>;<br/><a NAME="E16:40_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a></i>;<br/><a NAME="E17:40_1"/><b>then </b>
 <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">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T7" title="CLASSES1:th.7">CLASSES1:7</a></i>;<br/><a NAME="E18:40_1"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a></i>;<br/><a NAME="E19:40_1"/><b>then </b>
<font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E9:40_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="yellow19.html#E13:40_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="yellow19.html#E14:40_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E21:40_1"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:40"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_3"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span> <b>by </b><i><a class="txt" href="yellow19.html#E6:40_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="yellow19.html#E8:40_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="yellow19.html#E13:40_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="yellow_6.html#T12" title="YELLOW_6:th.12">YELLOW_6:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c12">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow19.html#E1:40_1_3"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/><a NAME="E4:40_1_3"/>
<font color="Maroon" title="c14">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:40_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="yellow19.html#E1:40_1_3"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>;<br/><a NAME="E5:40_1_3"/><b>then </b><i><font color="Green" title="E38">A16</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:40_1_3_1"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c15">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">V</font> )
 ;<br/><a NAME="E2:40_1_3_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><a NAME="E3:40_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:40_1_3_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><a NAME="E4:40_1_3_1"/><b>then </b>
<font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E21:40_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="waybel_9.html#D9" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span><b> such that </b><br/><a NAME="E6:40_1_3_1"/><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">V</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">a'</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c18">b'</font> = <font color="Maroon" title="c16">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <b>by </b><i><a class="txt" href="yellow19.html#E13:40_1"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/><a NAME="E8:40_1_3_1"/>
<font color="Maroon" title="c17">a'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">b'</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E13:40_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="yellow19.html#E6:40_1_3_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/><a NAME="E9:40_1_3_1"/><b>then </b>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c17">a'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c18">b'</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c17">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c18">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a>, <a class="ref" href="yellow_7.html#T1" title="YELLOW_7:th.1">YELLOW_7:1</a></i>;<br/><a NAME="E10:40_1_3_1"/><b>then </b>
( <font color="Maroon" title="c16">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">b</font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E5:40_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="yellow19.html#E8:40_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="yellow19.html#E13:40_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E11:40_1_3_1"/>
<font color="Maroon" title="c14">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E6:40_1_3_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:40_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E5:40_1_3"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:40_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E6:40_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/>


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