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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</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>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</span> ) ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:41"/>
<font color="Maroon">p</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="E3:41"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:41"/><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">p</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="E5:41"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</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="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/>
<a NAME="E6:41"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:41"><i><font color="Green">E26</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E7:41"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="waybel13.html#T1">WAYBEL13:1</a></i>;<br/>
<a NAME="E8:41"/>
 <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
 <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font>
 ;<br/>

<b>then </b><font color="Maroon">p</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E3:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel16.html#E4:41"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:41"/><b>then </b>
not  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:41"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E12:41"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:41"/>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">q</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E11:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_8.html#D2">WAYBEL_8:def 2</a></i>;<br/>
<b>then 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="E15:41"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 <b>and </b><br/><a NAME="E16:41"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">q</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E17:41"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> is <a href="waybel_3.html#V1">compact</a>
 ;<br/>
<a NAME="E18:41"/><i><font color="Green">E42</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>by </b><i><a class="ref" href="waybel16.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E19:41"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
;<br/>
<a NAME="E20:41"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</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="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E21:41"/>
not <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#E12:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel16.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/>
<a NAME="E22:41"/><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="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E23:41"/><b>then </b><i><font color="Green">E77</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="E24:41"/>
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="41_2"><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="E1:41_2"/><i><font color="Green">E78</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="E2:41_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:41_2"/>
<font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:41_2"/><b>then </b>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E15:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E5:41_2"/><b>then </b>
<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="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>hence </b><a NAME="E6:41_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:41"/><b>then </b>
<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/>
<a NAME="E26:41"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</span> ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><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>;<br/><b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</span> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:41_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:41_3"/>
 ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Maroon">a1</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="Olive">k</font></span>)</span> )
 ;<br/><b>thus </b><a NAME="E6:41_3"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</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="E28:41"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</span> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E29:41"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">u</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</span> ) ) </span>}</span>  holds <br/><font color="Maroon">p</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">u</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E30:41"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">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> &amp; <font color="Olive">x</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="Olive">k</font></span>)</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/>


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