<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V2">distributive</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_2.html#V2">meet-continuous</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:36"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">p</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">V</font> = 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#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> as  <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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> as  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 <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="yellow_7.html#T27">YELLOW_7:27</a></i>;<br/>
<b>take </b><font color="Maroon">k</font> =  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font>;<br/>

<a NAME="E5:36"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_2.html#V1">satisfying_MC</a>
 
<b>by </b><i>, <a class="ref" href="waybel_2.html#D7">WAYBEL_2:def 7</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E29</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">V</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V'</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E31</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E8:36"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E10:36"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<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="E1:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T15">YELLOW_4:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:36_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>
<a NAME="E6:36_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:36_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E8:36_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v</font> &amp; <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E9:36_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1"/>
<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">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font></span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T23">YELLOW_7:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel16.html#E1:36"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="yellow_7.html#T13">YELLOW_7:13</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="waybel16.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_2.html#D6">WAYBEL_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="lattice3.html#K8">~</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V'</font></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="waybel16.html#E1:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_7.html#T13">YELLOW_7:13</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel16.html#E5:36"><i><font color="Green">E28</font></i></a>, </i>
;<br/>
<a NAME="E12:36"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E2:36"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel16.html#E6:36"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E13:36"/><i><font color="Green">E78</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:36_3"/>
<font color="Maroon">b</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">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:36_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</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="E3:36_3"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</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="E15:36"/><b>then </b>
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E16:36"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E17:36"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E18:36"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_4"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:36_4"/><b>then </b>
<font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/>
<b>hence </b><a NAME="E3:36_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="waybel16.html#E8:36"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:36"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_5"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_1"><b>proof </b></a><div class="add">

<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="E1:36_5_1"/><i><font color="Green">E81</font></i>: 
<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>
 ;<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="E3:36_5_1"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E4:36_5_1"/><b>then </b>
not <font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T5" target="_self">Th5</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E5:36_5_1"/><b>then </b>
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<b>hence </b><a NAME="E6:36_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<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="E2:36_5"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</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="E4:36_5"/>
<font color="Maroon">k</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E5:36_5"/><b>then </b>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:36_5"/>
<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/>


</div><b>end;</b></div>
<a NAME="E20:36"/><b>then </b>
<font color="Maroon">k</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T2">WAYBEL_8:2</a></i>;<br/>
<b>hence </b><a NAME="E21:36"/>
<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>by </b><i><a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>

<a NAME="E22:36"/>
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="waybel16.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E23:36"/><b>then </b><i><font color="Green">E83</font></i>: 
<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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E24:36"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">y</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> or  not <font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:36_6"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">y</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>and </b><br/><a NAME="E2:36_6"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:36_6"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E4:36_6"/>
not <font color="Maroon">y</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/>
<a NAME="E5:36_6"/><b>then </b>
not <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="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E6:36_6"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E7:36_6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:36_6"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<b>hence </b><a NAME="E9:36_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:36"/>
<font color="Maroon">p</font> <a href="waybel_4.html#R3">is_maximal_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="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>


</div>
