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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:39"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattice6.html#V4" target="_self">completely-join-irreducible</a>
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lattice6.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:39"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:39_1"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<a NAME="E6:39_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:39_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="lattice6.html#E4:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E7:39_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>;<br/>
<a NAME="E8:39_1"/><i><font color="Green">E37</font></i>: 
 <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lattice6.html#T9" target="_self">Th9</a></i>;<br/>
<b>assume </b><a NAME="E9:39_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  )
 ;<br/>

<a NAME="E10:39_1"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:39_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="lattice6.html#E6:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lattice6.html#E7:39_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E11:39_1"/>
( <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E12:39_1"/><b>then </b>
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E10:39_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E13:39_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice6.html#E5:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E14:39_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E15:39_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E8:39_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E16:39_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp;  <a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice6.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E2:39"><i><font color="Green">E29</font></i></a>, <a class="txt" href="lattice6.html#E14:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice6.html#E15:39_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  is <a href="waybel_6.html#V3">join-irreducible</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D3">WAYBEL_6:def 3</a></i>;<br/>


</div>
