<?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="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>consider </b><font color="Maroon">S</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>
<b>let </b><font color="Maroon">N</font> be  <a href="waybel_0.html#V11">eventually-filtered</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">F</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:54"/><i><font color="Green">E26</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E30</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<i><font color="Green">E31</font></i>:  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font> = 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="waybel_9.html#D2">WAYBEL_9:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<a NAME="E6:54"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:54"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E8:54"/>
 <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:54_2"/>
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">ii</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:54_2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">ii</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T12">WAYBEL_0:12</a></i>;<br/>
<a NAME="E7:54_2_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T38">WAYBEL19:38</a></i>;<br/>
<a NAME="E8:54_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E9:54_2_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">ii</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 <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:54_2_1_1"/>
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ii</font>
 ;<br/>

<a NAME="E2:54_2_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:54_2_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:54_2_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:54_2_1"/><b>then </b>
 <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:54_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel19.html#T26">WAYBEL19:26</a></i>;<br/>
<b>hence </b><a NAME="E11:54_2_1"/>
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:54_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E5:54_2"/>
 <a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E6:54_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T2">YELLOW_2:2</a></i>;<br/>
<a NAME="E7:54_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T38">WAYBEL19:38</a></i>;<br/>
<a NAME="E8:54_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E9:54_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E10:54_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E7:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_9.html#T24">WAYBEL_9:24</a></i>;<br/>
<a NAME="E11:54_2"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:54_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E12:54_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E13:54_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel19.html#T30">WAYBEL19:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><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="E1:54_3"/><i><font color="Green">E54</font></i>: 
(  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="waybel21.html#E7:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1"><b>suppose </b></a><a NAME="E1:54_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E3:54_3_1_1"/>
<font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:54_3_1_1"/><b>then </b>
<font color="Maroon">a</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><a NAME="E5:54_3_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/><b>thus </b><a NAME="E7:54_3_1_1"/>
<font color="Maroon">N</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="54_3_1_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 <font color="Maroon">N</font>;<br/>

<a NAME="E1:54_3_1_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E6:54"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E2:54_3_1_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E3:54_3_1_1_1"/>
(  not <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> or <font color="Maroon">N</font> <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#E7:54"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_2"><b>suppose </b></a><a NAME="E1:54_3_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:54_3_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E4:54_3_1_2"/>
not  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:54"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/><a NAME="E5:54_3_1_2"/><b>then </b>
not <font color="Maroon">F</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</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="E7:54_3_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:54_3_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E10:54_3_1_2"/>
<font color="Maroon">N</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="54_3_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">ii</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:54_3_1_2_1"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">ii</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T12">WAYBEL_0:12</a></i>;<br/>
<b>take </b>
<font color="Maroon">ii</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 <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E4:54_3_1_2_1"/>
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ii</font>
 ;<br/>

<a NAME="E5:54_3_1_2_1"/><b>then </b>
( <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:54_3_1_2_1"/><b>then </b>
not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E7:54_3_1_2_1"/>
<font color="Maroon">N</font> <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="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:54"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="waybel19.html#T25">WAYBEL19:25</a></i>;<br/>
<b>hence </b><a NAME="E12:54"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div>
