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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:30"/>
<font color="Maroon">x</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:30"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">s</font> holds <br/><font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s</font>
 <b>and </b><br/><a NAME="E5:30"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:30"/><i><font color="Green">E28</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E7:30"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:30"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E9:30"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">q</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<a NAME="E10:30"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:30_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:30_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<a NAME="E4:30_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T1" target="_self">Th1</a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b>
<font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:30"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">q</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E12:30"/><b>then </b>
<font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<b>hence </b><a NAME="E13:30"/>
contradiction
 <b>by </b><i>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/>


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