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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><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>;<br/>



<b>set </b><font color="Maroon">N</font> =  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font>;<br/>
<a NAME="E1:44"/><i><font color="Green">E26</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">D</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">K</font> being    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E30</font></i>: 
 <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">A</font>
 ;<br/><a NAME="E2:44_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/><b>assume </b><a NAME="E3:44_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E4:44_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="waybel21.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E5:44_1"/><b>then </b>
<font color="Maroon">A</font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a> 
 <b>by </b><i><a class="ref" href="waybel19.html#T36">WAYBEL19:36</a></i>;<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="E7:44_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E8:44_1"/><i><font color="Green">E39</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">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:44"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E10:44_1"/>
 <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>;<br/>

<a NAME="E1:44_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>assume </b><a NAME="E2:44_1_1"/>
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:44_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:44_1_1"/>
<span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="txt" href="waybel21.html#E1:44_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel19.html#T25">WAYBEL19:25</a></i>;<br/>


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