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<b>let </b><font color="Maroon">T</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a> ;<br/>
<a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
 ex <font color="Olive">K</font> being   <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">K</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>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<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="E2:53_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>

<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="E4:53_1"/><i><font color="Green">E34</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/>
<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a><br/>

<b>assume </b><a NAME="E5:53_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E6:53_1"/><b>then </b><i><font color="Green">E36</font></i>: 
not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E9:53_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E10:53_1"/><i><font color="Green">E38</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a></i>;<br/>
<b>assume </b><a NAME="E11:53_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E12:53_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:53_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E10:53_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<b>hence </b><a NAME="E14:53_1"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:53"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="setfam_1.html#K5">union</a> <font color="Olive">F</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:53"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">A2</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E4:53"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>]
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:53"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="waybel19.html#S1" target="_self">WAYBEL19:sch 1</a>(, , , );</i><br/>


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