<?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="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">T</font> is <a href="waybel19.html#V2" target="_self">Lawson</a>
 ;<br/>

<a NAME="E2:62_1_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E3:62_1_1"/>
<font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#T27">URYSOHN1:27</a></i>;<br/>

<a NAME="E4:62_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">O1</font> =  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">O2</font> = <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/>
<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#T1" target="_self">Th1</a>, </i>;<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/>
<a NAME="E8:62_1_1"/><i><font color="Green">E34</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/>
<a NAME="E9:62_1_1"/>
for <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">K</font>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> holds <br/> ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">F</font> st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">K</font>;<br/>

<b>set </b><font color="Maroon">I2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>;<br/>
<a NAME="E1:62_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>assume </b><a NAME="E2:62_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:62_1_1_1"/><i><font color="Green">E38</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:62_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:62_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:62_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<a NAME="E2:62_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:62_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> , <font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel19.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font>, <font color="Maroon">J2</font> = <font color="Maroon">I2</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">z</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_1_2"/>
not <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:62_1_1_1_2"/><b>then </b>
<font color="Maroon">Y</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> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</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="E4:62_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E5:62_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:62_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <span class="p1">(<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>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><b>hence </b><a NAME="E7:62_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:62_1_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E12:62_1_1_1"/><b>then </b>
<font color="Maroon">Z</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="E13:62_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">Z</font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a>  &amp; <font color="Maroon">Z</font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i><a class="ref" href="waybel11.html#T15">WAYBEL11:15</a></i>;<br/>
<a NAME="E14:62_1_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">I2</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>then </b><font color="Maroon">z</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">J2</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">J2</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_0.html#T55">WAYBEL_0:55</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">S</font><b> such that </b><br/><a NAME="E17:62_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">J2</font>
 <b>and </b><br/><a NAME="E18:62_1_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">J2</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">Z</font>
 <b>by </b><i>, , <a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">J2</font><b> such that </b><br/><a NAME="E20:62_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">a</font>,<font color="Maroon">S</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">a</font>,<font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">AA</font> = <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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span> ;<br/>
<a NAME="E21:62_1_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:62_1_1_1_4"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  &amp; not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:62_1_1_1_4"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">i</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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:62_1_1_1_4"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:62_1_1_1_4"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:62_1_1_1_4"/><b>then </b>
 ex <font color="Olive">c'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c'</font> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">c'</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
;<br/>
<b>hence </b><a NAME="E7:62_1_1_1_4"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:62_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E23:62_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">G</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font> <b>by </b><i><a class="ref" href="waybel19.html#D2" target="_self">Def2</a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>let </b><font color="Maroon">u</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="E25:62_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E28:62_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:62_1_1_1"/>
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">u</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_1_1_1_6"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E2:62_1_1_1_6"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:62_1_1_1_6"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font>
 
;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:62_1_1_1_6"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E6:62_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K3">`</a>  <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">c</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> </span>}</span>  &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="waybel19.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E7:62_1_1_1_6"/><b>then </b>
( not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">v</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:62_1_1_1_6"/>
<font color="Maroon">v</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:62_1_1_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, , , <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E31:62_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:62_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T35">WAYBEL_7:35</a></i>;<br/>
<a NAME="E11:62_1_1"/><b>then </b>
<font color="Maroon">X</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:62_1_1"/>
<font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#T37">WAYBEL_3:37</a></i>;<br/>


</div>
