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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="waybel_0.html#V13">upper</a> <a href="waybel_6.html#V1">Open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">N</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="card_4.html#V1">countable</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>consider </b><font color="Maroon">O</font> being  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:56"/><i><font color="Green">E28</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="yellow_4.html#K4">"/\"</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E5:56"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E6:56"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i>, <a class="txt" href="waybel12.html#E11:56"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:56"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:56"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:56"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E11:56"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="waybel_6.html#T9">WAYBEL_6:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="waybel_6.html#V2">irreducible</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel_6.html#T13">WAYBEL_6:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E13:56"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>

<a NAME="E14:56"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T24">WAYBEL_0:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:56_1"/>
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:56"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><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> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">N</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_3.html#T7">YELLOW_3:7</a></i>;<br/>
<a NAME="E16:56"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>
<b>hence </b><a NAME="E17:56"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><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> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel12.html#T2" target="_self">Th2</a>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>


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