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

<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:100"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1">WAYBEL_6:def 1</a><br/>

<a NAME="E2:100"/>
<font color="Maroon">S</font> is <a href="waybel11.html#V1" target="_self">inaccessible_by_directed_joins</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" target="_self">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:100_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:100_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:100_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:100_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E7:100_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>
<a NAME="E8:100_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E9:100_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:100"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div>
