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

<b>let </b><font color="Maroon">N</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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">N</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><b>then reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">N</font> as  <a href="yellow_0.html#V1">lower-bounded</a> <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_9.html#NM1">TopLattice</a> ;<br/><a NAME="E3:48_1"/>
<font color="Maroon">L1</font> is <a href="waybel_2.html#V2">meet-continuous</a>
 ;<br/><b>hence </b><a NAME="E4:48_1"/>
( <font color="Maroon">N</font> is <a href="waybel_2.html#V2">meet-continuous</a> &amp; <font color="Maroon">N</font> is <a href="compts_1.html#V3">being_T2</a> )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:48"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">N</font> is <a href="waybel_2.html#V2">meet-continuous</a> &amp; <font color="Maroon">N</font> is <a href="compts_1.html#V3">being_T2</a> )
 ;<br/>

<b>thus </b><a NAME="E3:48"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a><br/>

<b>thus </b><a NAME="E4:48"/>
<font color="Maroon">N</font> is <a href="waybel_0.html#V24">up-complete</a>
 ;<br/>

<a NAME="E5:48"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st not <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> &amp; not <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<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">N</font>;<br/>
<a NAME="E3:48_2"/><i><font color="Green">E36</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">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">O</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">F</font></span>)</span></span>)</span> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>, <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">N</font> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="waybel19.html#T32">WAYBEL19:32</a></i>;<br/>
<a NAME="E5:48_2"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:48_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:48_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:48_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E10:48_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_8.html#T18">YELLOW_8:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">K</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:48_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E14:48_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">V1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="waybel30.html#E3:48"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">U2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>, <font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E16:48_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E17:48_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">U2</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E18:48_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:48_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E20:48_2"/>
<font color="Maroon">U2</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="txt" href="waybel30.html#E7:48_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<a NAME="E21:48_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T24">WAYBEL14:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">WU1</font> = <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E23:48_2"/>
<font color="Maroon">WU1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:48_2_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WU1</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:48_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:48_2_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WU1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:48_2_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, <a class="txt" href="waybel30.html#E1:48_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E5:48_2_1"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:48_2_1"/><b>then </b>
<font color="Maroon">WU1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<b>hence </b><a NAME="E7:48_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel30.html#E3:48_2"><i><font color="Green">E36</font></i></a>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:48_2"/><b>then </b>
<font color="Maroon">WU1</font> <a href="yellow_4.html#R4">is_coarser_than</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T20">WAYBEL12:20</a></i>;<br/>
<a NAME="E25:48_2"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">WU1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="yellow12.html#T28">YELLOW12:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">WU1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="txt" href="waybel30.html#E8:48_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E28:48_2"/>
<font color="Maroon">S</font> is <a href="waybel_2.html#V2">meet-continuous</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, , <a class="ref" href="yellow12.html#T14">YELLOW12:14</a></i>;<br/>
<a NAME="E29:48_2"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E30:48_2"/><i><font color="Green">E124</font></i>: 
(  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">WU1</font> &amp;  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E32:48_2"/><i><font color="Green">E125</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">S</font>
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T27">YELLOW_0:27</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
;<br/>
<a NAME="E34:48_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E3:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel30.html#E1:48_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel30.html#E3:48_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E35:48_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E36:48_2"/>
<font color="Maroon">U1</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T2" target="_self">Th2</a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E37:48_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="txt" href="waybel30.html#E9:48_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E38:48_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E39:48_2"/>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E40:48_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E41:48_2"/>
<font color="Maroon">N</font> is  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V2">Lawson</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E42:48_2"/><b>then </b>
<font color="Maroon">U2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T2" target="_self">Th2</a>, <a class="ref" href="waybel19.html#T37">WAYBEL19:37</a></i>;<br/>
<a NAME="E43:48_2"/><b>then </b>
<font color="Maroon">WU1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E44:48_2"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E45:48_2"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<a NAME="E46:48_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E48:48_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">x</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E50:48_2"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E52:48_2"/><i><font color="Green">E134</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="yellow12.html#T13">YELLOW12:13</a></i>;<br/>
<a NAME="E53:48_2"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM6">Semilattice</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">N</font>  st  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">N</font> st <br/>( <font color="Olive">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> &amp; not <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font> ) holds <br/> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">N</font> st <br/>( <font color="Olive">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> &amp; not <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>



<b>given </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><a NAME="E1:48_2_3"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:48_2_3"/><i><font color="Green">E147</font></i>: 
not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 ;<br/>

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

<b>thus </b><a NAME="E3:48_2_3"/>
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E4"><i><font color="Green">Lemma52</font></i></a></i>;<br/>

<b>assume </b><a NAME="E4:48_2_3"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:48_2_3"/>
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E6:48_2_3"/><b>then </b>
<font color="Maroon">u</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel30.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>
<b>hence </b><a NAME="E7:48_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">u1</font> = <font color="Maroon">u1</font>;<br/><b>thus </b><a NAME="E1:48_2_4"/>
<font color="Maroon">u1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="waybel30.html#E12:48_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel30.html#E3"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E2:48_2_4"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E12:48_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow12.html#T30">YELLOW12:30</a></i>;<br/><a NAME="E3:48_2_4"/><b>then </b>
not <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel30.html#E1:48_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel30.html#E3:48_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:48_2_4"/>
not <font color="Maroon">u1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">u2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E56:48_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">u2</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">u2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u2</font>
;<br/>

<b>thus </b><a NAME="E57:48_2"/>
( <font color="Maroon">u2</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">u2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="waybel30.html#E4"><i><font color="Green">Lemma52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
<font color="Maroon">N</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/>


</div>
