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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_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</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>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:37"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">z</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">z</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>assume </b><a NAME="E2:37"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">k1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">k1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:37"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E8:37"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E9:37"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>
<a NAME="E10:37"/>
not <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:37"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E11:37"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E12:37"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:37"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</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">k</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E14:37"/>
<font color="Maroon">k</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:37"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E15:37"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font> is  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T2">WAYBEL_8:2</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="E17:37"/><i><font color="Green">E41</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="E18:37"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="waybel_4.html#R3">is_maximal_in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="waybel16.html#E7:37"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_6.html#T9">WAYBEL_6:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<a NAME="E19:37"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E20:37"/>
<font color="Maroon">p</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 <b>by </b><i><a class="txt" href="waybel16.html#E8:37"><i><font color="Green">E31</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E21:37"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E22:37"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:37_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E5:37"><i><font color="Green">E28</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E3:37_1"/>
<font color="Maroon">p</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">k</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E8:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel16.html#E9:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>


</div>
