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<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">x</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="E1:28"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel16.html#K1" target="_self">Irr</a> <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:28"/>
<font color="Maroon">x1</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 
<b>by </b><i>, </i>;<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="E5:28"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E6:28"/><i><font color="Green">E28</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">x1</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="E7:28"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</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="E1:28_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:28_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E3:28_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 ;<br/><a NAME="E4:28_1"/>
( <font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E5:28_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x1</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="waybel16.html#T1" target="_self">Th1</a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E6:28_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E7:28_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">q</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="E8:28_1"/>
( <font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E9:28_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:28"/><b>then </b>
<font color="Maroon">x1</font> is <a href="waybel_6.html#V2">meet-irreducible</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#D2">WAYBEL_6:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:28"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K3">IRR</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_6.html#D4">WAYBEL_6:def 4</a></i>;<br/>


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